Daily Papers

In this paper, we develop rotation-equivariant neural networks for 4D panoptic segmentation. 4D panoptic segmentation is a recently established benchmark task for autonomous driving, which requires recognizing semantic classes and object instances on the road based on LiDAR scans, as well as assigning temporally consistent IDs to instances across time. We observe that the driving scenario is symmetric to rotations on the ground plane. Therefore, rotation-equivariance could provide better generalization and more robust feature learning. Specifically, we review the object instance clustering strategies, and restate the centerness-based approach and the offset-based approach as the prediction of invariant scalar fields and equivariant vector fields. Other sub-tasks are also unified from this perspective, and different invariant and equivariant layers are designed to facilitate their predictions. Through evaluation on the standard 4D panoptic segmentation benchmark of SemanticKITTI, we show that our equivariant models achieve higher accuracy with lower computational costs compared to their non-equivariant counterparts. Moreover, our method sets the new state-of-the-art performance and achieves 1st place on the SemanticKITTI 4D Panoptic Segmentation leaderboard.

The construction of a theory of quantum gravity is an outstanding problem that can benefit from better understanding the laws of nature that are expected to hold in regimes currently inaccessible to experiment. Such fundamental laws can be found by considering the classical counterparts of a quantum theory. For example, conservation laws in a quantum theory often stem from conservation laws of the corresponding classical theory. In order to construct such laws, this thesis is concerned with the interplay between symmetries and conservation laws of classical field theories and their application to asymptotically flat spacetimes. This work begins with an explanation of symmetries in field theories with a focus on variational symmetries and their associated conservation laws. Boundary conditions for general relativity are then formulated on three-dimensional asymptotically flat spacetimes at null infinity using the method of conformal completion. Conserved quantities related to asymptotic symmetry transformations are derived and their properties are studied. This is done in a manifestly coordinate independent manner. In a separate step a coordinate system is introduced, such that the results can be compared to existing literature. Next, asymptotically flat spacetimes which contain both future as well as past null infinity are considered. Asymptotic symmetries occurring at these disjoint regions of three-dimensional asymptotically flat spacetimes are linked and the corresponding conserved quantities are matched. Finally, it is shown how asymptotic symmetries lead to the notion of distinct Minkowski spaces that can be differentiated by conserved quantities.

Scalar fields are ubiquitous in theories of high-energy physics. In the context of cosmic inflation, this suggests the existence of spectator fields, which provide a subdominant source of energy density. We show that spectator fields boost the inflationary production of primordial black holes, with single-field ultra-slow roll evolution supplanted by a phase of evolution along the spectator direction, and primordial perturbations amplified by the resulting multifield dynamics. This generic mechanism is largely free from the severe fine-tuning that afflicts single-field inflationary PBH models.

Many learning algorithms have invariances: when their training data is transformed in certain ways, the function they learn transforms in a predictable manner. Here we formalize this notion using concepts from the mathematical field of category theory. The invariances that a supervised learning algorithm possesses are formalized by categories of predictor and target spaces, whose morphisms represent the algorithm's invariances, and an index category whose morphisms represent permutations of the training examples. An invariant learning algorithm is a natural transformation between two functors from the product of these categories to the category of sets, representing training datasets and learned functions respectively. We illustrate the framework by characterizing and contrasting the invariances of linear regression and ridge regression.

We present Manifold Diffusion Fields (MDF), an approach to learn generative models of continuous functions defined over Riemannian manifolds. Leveraging insights from spectral geometry analysis, we define an intrinsic coordinate system on the manifold via the eigen-functions of the Laplace-Beltrami Operator. MDF represents functions using an explicit parametrization formed by a set of multiple input-output pairs. Our approach allows to sample continuous functions on manifolds and is invariant with respect to rigid and isometric transformations of the manifold. Empirical results on several datasets and manifolds show that MDF can capture distributions of such functions with better diversity and fidelity than previous approaches.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

We consider holography for d-dimensional scale invariant but non-Lorentz invariant field theories, which do not admit the full Schrodinger symmetry group. We find new realizations of the corresponding (d+1)-dimensional gravity duals, engineered with a variety of matter Lagrangians, and their finite temperature generalizations. The thermodynamic properties of the finite temperature backgrounds are precisely those expected for anisotropic, scale invariant field theories. The brane and string theory realizations of such backgrounds are briefly discussed, along with their holographic interpretation in terms of marginal but non Lorentz invariant deformations of conformal field theories. We initiate discussion of holographic renormalization in these backgrounds, and note that such systematic renormalization is necessary to obtain the correct behavior of correlation functions.

Learning representations that capture the underlying data generating process is a key problem for data efficient and robust use of neural networks. One key property for robustness which the learned representation should capture and which recently received a lot of attention is described by the notion of invariance. In this work we provide a causal perspective and new algorithm for learning invariant representations. Empirically we show that this algorithm works well on a diverse set of tasks and in particular we observe state-of-the-art performance on domain generalization, where we are able to significantly boost the score of existing models.

We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.

We reassess the realistic discovery reach of Solar-System experiments for dark energy (DE) and dark matter (DM), making explicit the bridge from cosmology-level linear responses to local, screened residuals. In scalar-tensor frameworks with a universal conformal coupling A(phi) and chameleon/Vainshtein screening, we map cosmological responses {mu(z,k),Sigma(z,k)} inferred by DESI and Euclid to thin-shell or Vainshtein residuals in deep Solar potentials Phi_N. We emphasize a two-branch strategy. In a detection-first branch, a verified local anomaly -- an Einstein equivalence principle (EEP) violation, a Shapiro-delay signal with |gamma-1|simfewtimes 10^{-6}, an AU-scale Yukawa tail, or a ultralight DM (ULDM) line in clocks/atom interferometers in space (AIS) -- triggers a joint refit of cosmology and Solar-System data under a common microphysical parameterization {V(phi),A(phi)}. In a guardrail branch, Solar-System tests enforce constraints (EEP; PPN parameters gamma,beta; and dot G/G) and close unscreened or weakly screened corners indicated by cosmology. We forecast, per conjunction, |gamma-1|lesssim (2-5)times 10^{-6} (Ka-/X-band or optical Shapiro), eta_{EEP}sim (1--10)times 10^{-17} (drag-free AIS), |dot G/G|sim(3-5)times10^{-15},yr^{-1} (sub-mm-class LLR), a uniform ~2x tightening of AU-scale Yukawa/DM-density bounds, and (3-10)times improved ULDM-coupling reach from clocks. For a conformal benchmark, mu_{ lin,0}=0.10 implies chisimeq mu_{lin,0/2} and a Sun thin shell Delta R/Rlesssim (1/3chi)|gamma-1|/2=2.4times 10^{-3} at |gamma-1|=5times 10^{-6}; Vainshtein screening at 1 AU yields |gamma-1|lesssim 10^{-11}, naturally below near-term reach. We recommend a cost-effective guardrail+discovery portfolio with explicit triggers for escalation to dedicated missions.

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

Datasets often have their intrinsic symmetries, and particular deep-learning models called equivariant or invariant models have been developed to exploit these symmetries. However, if some or all of these symmetries are only approximate, which frequently happens in practice, these models may be suboptimal due to the architectural restrictions imposed on them. We tackle this issue of approximate symmetries in a setup where symmetries are mixed, i.e., they are symmetries of not single but multiple different types and the degree of approximation varies across these types. Instead of proposing a new architectural restriction as in most of the previous approaches, we present a regularizer-based method for building a model for a dataset with mixed approximate symmetries. The key component of our method is what we call equivariance regularizer for a given type of symmetries, which measures how much a model is equivariant with respect to the symmetries of the type. Our method is trained with these regularizers, one per each symmetry type, and the strength of the regularizers is automatically tuned during training, leading to the discovery of the approximation levels of some candidate symmetry types without explicit supervision. Using synthetic function approximation and motion forecasting tasks, we demonstrate that our method achieves better accuracy than prior approaches while discovering the approximate symmetry levels correctly.

Equivariant neural networks have shown improved performance, expressiveness and sample complexity on symmetrical domains. But for some specific symmetries, representations, and choice of coordinates, the most common point-wise activations, such as ReLU, are not equivariant, hence they cannot be employed in the design of equivariant neural networks. The theorem we present in this paper describes all possible combinations of finite-dimensional representations, choice of coordinates and point-wise activations to obtain an exactly equivariant layer, generalizing and strengthening existing characterizations. Notable cases of practical relevance are discussed as corollaries. Indeed, we prove that rotation-equivariant networks can only be invariant, as it happens for any network which is equivariant with respect to connected compact groups. Then, we discuss implications of our findings when applied to important instances of exactly equivariant networks. First, we completely characterize permutation equivariant networks such as Invariant Graph Networks with point-wise nonlinearities and their geometric counterparts, highlighting a plethora of models whose expressive power and performance are still unknown. Second, we show that feature spaces of disentangled steerable convolutional neural networks are trivial representations.

Many machine learning tasks involve learning functions that are known to be invariant or equivariant to certain symmetries of the input data. However, it is often challenging to design neural network architectures that respect these symmetries while being expressive and computationally efficient. For example, Euclidean motion invariant/equivariant graph or point cloud neural networks. We introduce Frame Averaging (FA), a general purpose and systematic framework for adapting known (backbone) architectures to become invariant or equivariant to new symmetry types. Our framework builds on the well known group averaging operator that guarantees invariance or equivariance but is intractable. In contrast, we observe that for many important classes of symmetries, this operator can be replaced with an averaging operator over a small subset of the group elements, called a frame. We show that averaging over a frame guarantees exact invariance or equivariance while often being much simpler to compute than averaging over the entire group. Furthermore, we prove that FA-based models have maximal expressive power in a broad setting and in general preserve the expressive power of their backbone architectures. Using frame averaging, we propose a new class of universal Graph Neural Networks (GNNs), universal Euclidean motion invariant point cloud networks, and Euclidean motion invariant Message Passing (MP) GNNs. We demonstrate the practical effectiveness of FA on several applications including point cloud normal estimation, beyond 2-WL graph separation, and n-body dynamics prediction, achieving state-of-the-art results in all of these benchmarks.

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.

We prove existence of weak and strong solutions and uniqueness for a viscous dyadic model driven by additive white noise in time using a path-wise approach. Existence of invariant measures also established and a simple balance relation among the mean rates of energy injection, dissipation and flux is derived and we investigate the asymptotic exponents zeta_{p} of the p-order structure functions.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

In this work, we embark on the thermodynamic investigation concerning a family of primary charged black holes within the context of shift and parity symmetric Beyond Horndeski gravity. Employing the Euclidean approach, we derive the functional expression for the free energy and derive the first thermodynamic law, offering a methodology to address the challenge of extracting the thermal quantities in shift-symmetric scalar tensor theories characterized by linear time dependence in the scalar field. Following the formal analysis, we provide some illustrative examples focusing on the thermal evaporation of these fascinating objects.

We propose a simple interpretation of the gammagamma excesses reported by both CMS and ATLAS groups at 95 GeV together with the LEP excess in the Zbb channel around the same mass in terms of a neutral scalar field in the minimal left-right symmetric model (LRSM). We point out that the scalar field which implements the seesaw mechanism for neutrino masses has all the right properties to explain these observations, without introducing any extra scalar fields. The key point is that this scalar particle is hardly constrained because it couples only to heavy right-handed particles. As a result, the diphoton decay mode receives contributions from both mixing with the Standard Model (SM) Higgs and the heavy charged bosons in the LRSM, depending on the SU(2)_Rtimes U(1)_{B-L} symmetry breaking scale v_R. The complete allowed parameter space for explaining the 95 GeV excesses in this model can be probed with the high-precision measurements of the SM Higgs mixing with other scalars at the high-luminosity LHC and future Higgs factories.

We allow the Higgs field Phi to interact with a dilaton field chi of the background spacetime via the coupling chi^2,Phi^daggerPhi. Upon spontaneous gauge symmetry breaking, the Higgs VEV becomes proportional to chi. While traditionally this linkage is employed to make the Planck mass and particle masses dependent on chi, we present an textit alternative mechanism: the Higgs VEV will be used to construct Planck's constant hbar and speed of light c. Specifically, each open set vicinity of a given point x^* on the spacetime manifold is equipped with a replica of the Glashow-Weinberg-Salam action operating with its own effective values of hbar_* and c_* per hbar_*proptochi^{-1/2}(x^*) and c_*proptochi^{1/2}(x^*), causing these ``fundamental constants'' to vary alongside the dynamical field chi. Moreover, in each open set around x^*, the prevailing value chi(x^*) determines the length and time scales for physical processes occurring in this region as lproptochi^{-1}(x^*) and tauproptochi^{-3/2}(x^*). This leads to an textit anisotropic relation tau^{-1}propto l^{-3/2} between the rate of clocks and the length of rods, resulting in a distinct set of novel physical phenomena. For late-time cosmology, the variation of c along the trajectory of light waves from distant supernovae towards the Earth-based observer necessitates modifications to the Lema\^itre redshift relation and the Hubble law. These modifications are capable of: (1) Accounting for the Pantheon Catalog of SNeIa through a declining speed of light in an expanding Einstein--de Sitter universe, thus avoiding the need for dark energy; (2) Revitalizing Blanchard-Douspis-Rowan-Robinson-Sarkar's CMB power spectrum analysis that bypassed dark energy [A&A 412, 35 (2003)]; and (3) Resolving the H_0 tension without requiring a dynamical dark energy component.

We find candidate macroscopic gravity duals for scale-invariant but non-Lorentz invariant fixed points, which do not have particle number as a conserved quantity. We compute two-point correlation functions which exhibit novel behavior relative to their AdS counterparts, and find holographic renormalization group flows to conformal field theories. Our theories are characterized by a dynamical critical exponent z, which governs the anisotropy between spatial and temporal scaling t to lambda^z t, x to lambda x; we focus on the case with z=2. Such theories describe multicritical points in certain magnetic materials and liquid crystals, and have been shown to arise at quantum critical points in toy models of the cuprate superconductors. This work can be considered a small step towards making useful dual descriptions of such critical points.

The Weak Gravity Conjecture has recently been re-formulated in terms of a particle with non-negative self-binding energy. Because of the dual conformal field theory (CFT) formulation in the anti-de Sitter space the conformal dimension Delta (Q) of the lowest-dimension operator with charge Q under some global U(1) symmetry must be a convex function of Q. This property has been conjectured to hold for any (unitary) conformal field theory and generalized to larger global symmetry groups. Here we refine and further test the convex charge conjecture via semiclassical computations for fixed charge sectors of different theories in different dimensions. We analyze the convexity properties of the leading and next-to-leading order terms stemming from the semiclassical computation, de facto, extending previous tests beyond the leading perturbative contributions and to arbitrary charges. In particular, the leading contribution is sufficient to test convexity in the semiclassical computations. We also consider intriguing cases in which the models feature a transition from real to complex conformal dimensions either as a function of the charge or number of matter fields. As a relevant example of the first kind, we investigate the O(N) model in 4+epsilon dimensions. As an example of the second type we consider the U(N)times U(M) model in 4-epsilon dimensions. Both models display a rich dynamics where, by changing the number of matter fields and/or charge, one can achieve dramatically different physical regimes. We discover that whenever a complex conformal dimension appears, the real part satisfies the convexity property.

We present the analytic evaluation of the gravitational energy and of the angular momentum flux with tidal effects for inspiraling compact binaries, at next-to-next-to-leading post-Newtoian (2PN) order, within the effective field theory diagrammatic approach. We first compute the stress-energy tensor for a binary system, that requires the evaluation of two-point Feynman integrals, up to two loops. Then, we extract the multipole moments of the system, which we present for generic orbits in center-of-mass coordinates, and which are needed for the evaluation of the total gravitational energy and the angular momentum flux, for generic orbits. Finally, we provide the expression of gauge invariant quantities such as the fluxes, and the mode amplitudes and phase of the emitted gravitational wave, for circular orbits. Our findings are useful to update earlier theoretical studies as well as related phenomenological analyses, and waveform models

The dynamics of Einstein-conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman-Robertson--Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (`Small Bang'), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (`Big Bang'). Thus, in the present EccH model there is a new kind of physical spacetime singularity (`Small Bang'). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout-Englert-Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

We find and analyse solutions of Einstein's equations in arbitrary d dimensions and in the presence of a scalar field with a Liouville potential coupled to a Maxwell field. We consider spacetimes of cylindrical symmetry or again subspaces of dimension d-2 with constant curvature and analyse in detail the field equations and manifest their symmetries. The field equations of the full system are shown to reduce to a single or couple of ODE's which can be used to solve analytically or numerically the theory for the symmetry at hand. Further solutions can also be generated by a solution generating technique akin to the EM duality in the absence of a cosmological constant. We then find and analyse explicit solutions including black holes and gravitating solitons for the case of four dimensional relativity and the higher-dimensional oxydised 5-dimensional spacetime. The general solution is obtained for a certain relation between couplings in the case of cylindrical symmetry.

In the 3-3-1 model with right-handed neutrinos, three triplets of scalars engender the correct sequence of symmetry breaking, SU(3)_C times SU(3)_L times U(1)_X rightarrow SU(3)_C times SU(2)_L times U(1)_Y rightarrow SU(3)_C times U(1)_{EM}, generating mass for all fermions, except neutrinos. Tiny neutrino masses may be achieved by adding one sextet of scalars to the original scalar content. As consequence, it emerges a very complex scalar sector, involving terms that violate lepton number explicitly, too. The main obstacle to the development of the phenomenology of such scenario is the knowledge of its spectrum of scalars since, now, there are 15 massive scalar particles on it. The proposal of this work is to do an exhaustive analysis of such scalar sector with lepton number being explicitly violated at low, electroweak and high energy scales by means of trilinear terms in the potential. The first case can be addressed analytically and, as a nice result, we have observed that the scalar content of such case is split into two categories: One belonging to the 331 energy scale and the other belonging to the EWSB energy scale, with the last recovering the well known THDM+triplet. For the other cases, the scalar sector can be addressed only numerically. Hence, we proposed a very general approach for the numerical study of the potential, avoiding simplifications that can make us reach conclusions without foundation. We show that, in the case of lepton number being explicitly violated at electroweak scale, it is possible to recover the same physics of the THDM+triplet, as the previous case. Among all the possibilities, we call the attention to one special case which generates the 3HDM+triplet scenario. For the last case, when lepton number is violated at high energy scale, the sextet become very massive and decouples from the original scalar content of the 3-3-1 model.

We discuss relevant scalar deformations of a holographic theory with a compact boundary. An example of such a theory would be the global AdS_4 with its spatially compact boundary S^2. To introduce a relevant deformation, we choose to turn on a time-independent and spatially homogeneous non-normalizable scalar operator with m^2 = -2. The finite size of a compact boundary cuts down the RG flow at a finite length scale leading to an incomplete RG flow to IR. We discuss a version of {\it incomplete} C-theorem and an {\it incomplete} attractor like mechanism. We discuss the implication of our results for entanglement entropy and geometric quantities like scalar curvature, volume and mass scale of fundamental excitation of the how these quantities increase or decrease (often monotonically) with the strength of the deformation. Thermal physics of a holographic theory defined on a compact boundary is more interesting than its non-compact counterpart. It is well known that with a compact boundary, there is a possibility of a first order Hawking-Page transition dual to a de-confinement phase transition. From a gravity perspective, a relevant deformation dumps negative energy inside the bulk, increasing the effective cosmological constant (Lambda) of the AdS. Dumping more negative energy in the bulk would make the HP transition harder and the corresponding HP transition temperature would increase. However, we have found the size of the BH at the transition temperature decreases.

In this paper we study quasinormal modes and absorption cross sections for the (1+3)-dimensional Bardeen black hole surrounded by perfect fluid dark matter. Studies of the massless scalar field is already done in Sun:2023slzl. Hence, in this paper we will focus on the massive scalar field perturbations and massless Dirac field perturbations. To compute the quasinormal modes we use the semi-analytical 3rd-order WKB method, which has been shown to be one of the best approaches when the effective potential is adequate and when n < ell and n < lambda. We have also utilized the P\"oschl-Teller method to compare the valus obtained using the WKB approach. We have computed quasinormal frequencies by varying various parameters of the theory such as the mass of the scalar field mu, dark matter parameter alpha and the magnetic charge g. We have summarized our solutions in tables and figures for clarity. As for the absorption cross section, we used third order WKB approach to compute reflection, transmission coefficients and partial absorption cross sections. Graphs are presented to demonstrate the behavior of the above quantities when the dark matter parameter and mass of the massive scalar field are varied.

Recent progress has been made towards learning invariant or equivariant representations with self-supervised learning. While invariant methods are evaluated on large scale datasets, equivariant ones are evaluated in smaller, more controlled, settings. We aim at bridging the gap between the two in order to learn more diverse representations that are suitable for a wide range of tasks. We start by introducing a dataset called 3DIEBench, consisting of renderings from 3D models over 55 classes and more than 2.5 million images where we have full control on the transformations applied to the objects. We further introduce a predictor architecture based on hypernetworks to learn equivariant representations with no possible collapse to invariance. We introduce SIE (Split Invariant-Equivariant) which combines the hypernetwork-based predictor with representations split in two parts, one invariant, the other equivariant, to learn richer representations. We demonstrate significant performance gains over existing methods on equivariance related tasks from both a qualitative and quantitative point of view. We further analyze our introduced predictor and show how it steers the learned latent space. We hope that both our introduced dataset and approach will enable learning richer representations without supervision in more complex scenarios. Code and data are available at https://github.com/facebookresearch/SIE.

We investigate thermodynamics of static and spherically symmetric black holes (BHs) in the Horndeski theories. Because of the presence of the higher-derivative interactions and the nonminimal derivative couplings of the scalar field, the standard Wald entropy formula may not be directly applicable. Hence, following the original formulation by Iyer and Wald, we obtain the differentials of the BH entropy and the total mass of the system in the Horndeski theories, which lead to the first-law of thermodynamics via the conservation of the Hamiltonian. Our formulation covers the case of the static and spherically symmetric BH solutions with the static scalar field and those with the linearly time-dependent scalar field in the shift-symmetric Horndeski theories. We then apply our results to explicit BH solutions in the Horndeski theories. In the case of the conventional scalar-tensor theories and the Einstein-scalar-Gauss-Bonnet theories, we recover the BH entropy obtained by the Wald entropy formula. In the shift-symmetric theories, in the case of the BH solutions with the static scalar field we show that the BH entropy follows the ordinary area law even in the presence of the nontrivial profile of the scalar field. On the other hand, in the case of the BH solutions where the scalar field linearly depends on time, i.e., the stealth Schwarzschild and Schwarzschild-(anti-) de Sitter solutions, the BH entropy also depends on the profile of the scalar field. By use of the entropy, we find that there exists some range of the parameters in which Schwarzschild-(AdS) BH with non-trivial scalar field is thermodynamically stable than Schwarzschild-(AdS) BH without scalar field in general relativity.

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Let mathbb H be the finite direct sums of H^2(mathbb D). In this paper, we give a characterization of the closed subspaces of mathbb H which are invariant under the shift, thus obtaining a concrete Beurling-type theorem for the finite index shift. This characterization presents any such a subspace as the finite intersection, up to an inner function, of pre-images of a closed shift-invariant subspace of H^2(mathbb D) under ``determinantal operators'' from mathbb H to H^2(mathbb D), that is, continuous linear operators which intertwine the shifts and appear as determinants of matrices with entries given by bounded holomorphic functions. With simple algebraic manipulations we provide a direct proof that every invariant closed subspace of codimension at least two sits into a non-trivial closed invariant subspace. As a consequence every bounded linear operator with finite defect has a nontrivial closed invariant subspace.

Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena. The resulting representations are then combined with global parameters and directly decoded into continuous output fields, thus efficiently modeling the coupling between geometry and physics. By leveraging the inductive biases of locality and translation invariance, our approach is able to capture fine-scale physical features as well as complex shape variations, thereby enhancing generalization and physical compliance. Extensive experiments on a high-fidelity aerodynamic dataset, a hyper-elastic material benchmark, and multi-element airfoil geometries, demonstrate that the proposed model achieves superior or competitive performance compared to state-of-the-art graph based, operator learning, and neural field methods. Notably, our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes while maintaining high accuracy on full-scale discretizations.

Hypernetworks, neural networks that predict the parameters of another neural network, are powerful models that have been successfully used in diverse applications from image generation to multi-task learning. Unfortunately, existing hypernetworks are often challenging to train. Training typically converges far more slowly than for non-hypernetwork models, and the rate of convergence can be very sensitive to hyperparameter choices. In this work, we identify a fundamental and previously unidentified problem that contributes to the challenge of training hypernetworks: a magnitude proportionality between the inputs and outputs of the hypernetwork. We demonstrate both analytically and empirically that this can lead to unstable optimization, thereby slowing down convergence, and sometimes even preventing any learning. We present a simple solution to this problem using a revised hypernetwork formulation that we call Magnitude Invariant Parametrizations (MIP). We demonstrate the proposed solution on several hypernetwork tasks, where it consistently stabilizes training and achieves faster convergence. Furthermore, we perform a comprehensive ablation study including choices of activation function, normalization strategies, input dimensionality, and hypernetwork architecture; and find that MIP improves training in all scenarios. We provide easy-to-use code that can turn existing networks into MIP-based hypernetworks.

Machine learning models will often fail when deployed in an environment with a data distribution that is different than the training distribution. When multiple environments are available during training, many methods exist that learn representations which are invariant across the different distributions, with the hope that these representations will be transportable to unseen domains. In this work, we present a nonparametric strategy for learning invariant representations based on the recently-proposed Nadaraya-Watson (NW) head. The NW head makes a prediction by comparing the learned representations of the query to the elements of a support set that consists of labeled data. We demonstrate that by manipulating the support set, one can encode different causal assumptions. In particular, restricting the support set to a single environment encourages the model to learn invariant features that do not depend on the environment. We present a causally-motivated setup for our modeling and training strategy and validate on three challenging real-world domain generalization tasks in computer vision.

In state-of-the-art self-supervised learning (SSL) pre-training produces semantically good representations by encouraging them to be invariant under meaningful transformations prescribed from human knowledge. In fact, the property of invariance is a trivial instance of a broader class called equivariance, which can be intuitively understood as the property that representations transform according to the way the inputs transform. Here, we show that rather than using only invariance, pre-training that encourages non-trivial equivariance to some transformations, while maintaining invariance to other transformations, can be used to improve the semantic quality of representations. Specifically, we extend popular SSL methods to a more general framework which we name Equivariant Self-Supervised Learning (E-SSL). In E-SSL, a simple additional pre-training objective encourages equivariance by predicting the transformations applied to the input. We demonstrate E-SSL's effectiveness empirically on several popular computer vision benchmarks, e.g. improving SimCLR to 72.5% linear probe accuracy on ImageNet. Furthermore, we demonstrate usefulness of E-SSL for applications beyond computer vision; in particular, we show its utility on regression problems in photonics science. Our code, datasets and pre-trained models are available at https://github.com/rdangovs/essl to aid further research in E-SSL.

We introduce Einstein Fields, a neural representation that is designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights. By modeling the metric, which is the core tensor field of general relativity, Einstein Fields enable the derivation of physical quantities via automatic differentiation. However, unlike conventional neural fields (e.g., signed distance, occupancy, or radiance fields), Einstein Fields are Neural Tensor Fields with the key difference that when encoding the spacetime geometry of general relativity into neural field representations, dynamics emerge naturally as a byproduct. Einstein Fields show remarkable potential, including continuum modeling of 4D spacetime, mesh-agnosticity, storage efficiency, derivative accuracy, and ease of use. We address these challenges across several canonical test beds of general relativity and release an open source JAX-based library, paving the way for more scalable and expressive approaches to numerical relativity. Code is made available at https://github.com/AndreiB137/EinFields

Systems of interacting objects often evolve under the influence of field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions -- which are SE(n) equivariant and depend on relative states -- from external global field effects -- which depend on absolute states. We model interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

In our prior work toward Bartnik's static vacuum extension conjecture for near Euclidean boundary data, we establish a sufficient condition, called static regular, and confirm large classes of boundary hypersurfaces are static regular. In this note, we further improve some of those prior results. Specifically, we show that any hypersurface in an open and dense subfamily of a certain general smooth one-sided family of hypersurfaces (not necessarily a foliation) is static regular. The proof uses some of our new arguments motivated from studying the conjecture for boundary data near an arbitrary static vacuum metric.

We study black branes carrying both electric and magnetic charges in Einstein-Maxwell theory coupled to a dilaton-axion in asymptotically anti de Sitter space. After reviewing and extending earlier results for the case of electrically charged branes, we characterise the thermodynamics of magnetically charged branes. We then focus on dyonic branes in theories which enjoy an SL(2,R) electric-magnetic duality. Using SL(2,R), we are able to generate solutions with arbitrary charges starting with the electrically charged solution, and also calculate transport coefficients. These solutions all exhibit a Lifshitz-like near-horizon geometry. The system behaves as expected for a charged fluid in a magnetic field, with non-vanishing Hall conductance and vanishing DC longitudinal conductivity at low temperatures. Its response is characterised by a cyclotron resonance at a frequency proportional to the magnetic field, for small magnetic fields. Interestingly, the DC Hall conductance is related to the attractor value of the axion. We also study the attractor flows of the dilaton-axion, both in cases with and without an additional modular-invariant scalar potential. The flows exhibit intricate behaviour related to the duality symmetry. Finally, we briefly discuss attractor flows in more general dilaton-axion theories which do not enjoy SL(2,R) symmetry.

We introduce Invariant Risk Minimization (IRM), a learning paradigm to estimate invariant correlations across multiple training distributions. To achieve this goal, IRM learns a data representation such that the optimal classifier, on top of that data representation, matches for all training distributions. Through theory and experiments, we show how the invariances learned by IRM relate to the causal structures governing the data and enable out-of-distribution generalization.

We propose a general theory of the Open Gromov-Witten invariant on Calabi-Yau three-folds. We introduce the moduli space of multi-curves and show how it leads to invariants. Our construction is based on an idea of Witten. In the special case that each connected component of the Lagrangian submanifold has the rational homology of a sphere we define rational numbers F_{g,h} for each genus g and h boundary components.

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.

Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics-informed ROM (Φ-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, Φ-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across various PDE solvers and highlight its broad applicability by providing an open-source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi-rom.github.io.

We reformulate the Landau analysis of Feynman integrals with the aim of advancing the state of the art in modern particle-physics computations. We contribute new algorithms for computing Landau singularities, using tools from polyhedral geometry and symbolic/numerical elimination. Inspired by the work of Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we define the principal Landau determinant of a Feynman diagram. We illustrate with a number of examples that this algebraic formalism allows to compute many components of the Landau singular locus. We adapt the GKZ framework by carefully specializing Euler integrals to Feynman integrals. For instance, ultraviolet and infrared singularities are detected as irreducible components of an incidence variety, which project dominantly to the kinematic space. We compute principal Landau determinants for the infinite families of one-loop and banana diagrams with different mass configurations, and for a range of cutting-edge Standard Model processes. Our algorithms build on the Julia package Landau.jl and are implemented in the new open-source package PLD.jl available at https://mathrepo.mis.mpg.de/PLD/.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2^nd-order optimizers for deep learning in low precision settings. Code: https://github.com/yorkerlin/StructuredNGD-DL

We obtain (small-parameter) well-posedness for the (space-time periodic) Phi^4 equation in the full subcritical regime in the context of regularity structures based on multi-indices. As opposed to Hairer's more extrinsic tree-based setting, due to the intrinsic description encoded by multi-indices, it is not possible to obtain a solution theory via the standard fixed-point argument. Instead, we develop a more intrinsic approach for existence using a variant of the continuity method from classical PDE theory based on a priori estimates for a new `robust' formulation of the equation. This formulation also allows us to obtain uniqueness of solutions and continuity of the solution map in the model norm even at the limit of vanishing regularisation scale. Since our proof relies on the structure of the nonlinearity in only a mild way, we expect the same ideas to be sufficient to treat a more general class of equations.

It has been observed that representations learned by distinct neural networks conceal structural similarities when the models are trained under similar inductive biases. From a geometric perspective, identifying the classes of transformations and the related invariances that connect these representations is fundamental to unlocking applications, such as merging, stitching, and reusing different neural modules. However, estimating task-specific transformations a priori can be challenging and expensive due to several factors (e.g., weights initialization, training hyperparameters, or data modality). To this end, we introduce a versatile method to directly incorporate a set of invariances into the representations, constructing a product space of invariant components on top of the latent representations without requiring prior knowledge about the optimal invariance to infuse. We validate our solution on classification and reconstruction tasks, observing consistent latent similarity and downstream performance improvements in a zero-shot stitching setting. The experimental analysis comprises three modalities (vision, text, and graphs), twelve pretrained foundational models, nine benchmarks, and several architectures trained from scratch.

In the present work, we study the dynamical evolution of an homogeneous and anisotropic, noncommutative (NC) Bianchi I (BI) model coupled to a radiation perfect fluid. Our first motivation is determining if the present model tends to an homogeneous and isotropic NC Friedmann-Robertson-Walker (FRW) model, during its evolution. In order to simplify our task, we use the Misner parametrization of the BI metric. In terms of that parametrization the BI metric has three metric functions: the scale factor a(t) and the two parameters beta_pm (t), which measure the spatial anisotropy of the model. Our second motivation is trying to describe the present accelerated expansion of the universe using noncommutativity (NCTY). The NCTY is introduced by two nontrivial Poisson brackets between some geometrical as well as matter variables of the model. We recover the description in terms of commutative variables by introducing some variables transformations that depend on the NC parameter. Using those variables transformations, we rewrite the total NC Hamiltonian of the model in terms of commutative variables. From the resulting Hamiltonian, we obtain the dynamical equations for a generic perfect fluid. In order to solve these equations, we restrict our attention to a model where the perfect fluid is radiation. We solve, numerically, these equations and compare the NC solutions to the corresponding commutative ones. The comparison shows that the NC model may be considered as a possible candidate for describing the accelerated expansion of the universe. Finally, we obtain estimates for the NC parameter and compare the main results of the NC BI model coupled to radiation with the same NC BI model coupled to other perfect fluids. As our main result, we show that the solutions, after some time, produce an isotropic universe.

In the present paper we study the singularity structure of the exterior field of neutron stars with the aid of the four-parameter exact solution of the Einstein-Maxwell equations. The complete analysis of this problem in the generic case becomes possible due to the implementation of the novel analytical approach to the resolution of the singularity condition, and it shows the absence of the ring singularities off the symmetry axis in the positive mass case, as well as the possibility of the removal of the ring singularity by a strong magnetic field in the negative mass case. The solution takes an extraordinarily simple form in the equatorial plane, very similar to that of the Kerr solution, which makes it most suitable for astrophysical applications as the simplest model of a rotating magnetized deformed mass. It also provides a nontrivial example confirming a recent claim that the varphi component of the electromagnetic four-potential has features inconsistent with the intrinsic properties of the electrovac metric, while the magnetic field is represented correctly by the t component of the dual electromagnetic four-potential.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

Learning to predict agent motions with relationship reasoning is important for many applications. In motion prediction tasks, maintaining motion equivariance under Euclidean geometric transformations and invariance of agent interaction is a critical and fundamental principle. However, such equivariance and invariance properties are overlooked by most existing methods. To fill this gap, we propose EqMotion, an efficient equivariant motion prediction model with invariant interaction reasoning. To achieve motion equivariance, we propose an equivariant geometric feature learning module to learn a Euclidean transformable feature through dedicated designs of equivariant operations. To reason agent's interactions, we propose an invariant interaction reasoning module to achieve a more stable interaction modeling. To further promote more comprehensive motion features, we propose an invariant pattern feature learning module to learn an invariant pattern feature, which cooperates with the equivariant geometric feature to enhance network expressiveness. We conduct experiments for the proposed model on four distinct scenarios: particle dynamics, molecule dynamics, human skeleton motion prediction and pedestrian trajectory prediction. Experimental results show that our method is not only generally applicable, but also achieves state-of-the-art prediction performances on all the four tasks, improving by 24.0/30.1/8.6/9.2%. Code is available at https://github.com/MediaBrain-SJTU/EqMotion.

Geometric deep learning, i.e., designing neural networks to handle the ubiquitous geometric data such as point clouds and graphs, have achieved great successes in the last decade. One critical inductive bias is that the model can maintain invariance towards various transformations such as translation, rotation, and scaling. The existing graph neural network (GNN) approaches can only maintain permutation-invariance, failing to guarantee invariance with respect to other transformations. Besides GNNs, other works design sophisticated transformation-invariant layers, which are computationally expensive and difficult to be extended. To solve this problem, we revisit why the existing neural networks cannot maintain transformation invariance when handling geometric data. Our findings show that transformation-invariant and distance-preserving initial representations are sufficient to achieve transformation invariance rather than needing sophisticated neural layer designs. Motivated by these findings, we propose Transformation Invariant Neural Networks (TinvNN), a straightforward and general framework for geometric data. Specifically, we realize transformation-invariant and distance-preserving initial point representations by modifying multi-dimensional scaling before feeding the representations into neural networks. We prove that TinvNN can strictly guarantee transformation invariance, being general and flexible enough to be combined with the existing neural networks. Extensive experimental results on point cloud analysis and combinatorial optimization demonstrate the effectiveness and general applicability of our proposed method. Based on the experimental results, we advocate that TinvNN should be considered a new starting point and an essential baseline for further studies of transformation-invariant geometric deep learning.

Visualizing features in deep neural networks (DNNs) can help understanding their computations. Many previous studies aimed to visualize the selectivity of individual units by finding meaningful images that maximize their activation. However, comparably little attention has been paid to visualizing to what image transformations units in DNNs are invariant. Here we propose a method to discover invariances in the responses of hidden layer units of deep neural networks. Our approach is based on simultaneously searching for a batch of images that strongly activate a unit while at the same time being as distinct from each other as possible. We find that even early convolutional layers in VGG-19 exhibit various forms of response invariance: near-perfect phase invariance in some units and invariance to local diffeomorphic transformations in others. At the same time, we uncover representational differences with ResNet-50 in its corresponding layers. We conclude that invariance transformations are a major computational component learned by DNNs and we provide a systematic method to study them.

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting G-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the Pin(p,q,r) group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable geometric templates that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

It has been shown that a gravitational dual to a superconductor can be obtained by coupling anti-de Sitter gravity to a Maxwell field and charged scalar. We review our earlier analysis of this theory and extend it in two directions. First, we consider all values for the charge of the scalar field. Away from the large charge limit, backreaction on the spacetime metric is important. While the qualitative behaviour of the dual superconductor is found to be similar for all charges, in the limit of arbitrarily small charge a new type of black hole instability is found. We go on to add a perpendicular magnetic field B and obtain the London equation and magnetic penetration depth. We show that these holographic superconductors are Type II, i.e., starting in a normal phase at large B and low temperatures, they develop superconducting droplets as B is reduced.

The classical Seiberg-Witten equations in dimensions three and four admit a natural generalization within a unified framework known as the generalized Seiberg-Witten (GSW) equations, which encompasses many important equations in gauge theory. This article proves that the averaged L^2-norm of any spinor with non-constant pointwise norm in the GSW equations on mathbb R^4 and mathbb R^3, measured over large-radius spheres, grows faster than a power of the radius, under a suitable curvature decay assumption. Separately, it is shown that if the Yang-Mills-Higgs energy of any solution of these equations is finite, then the pointwise norm of the spinor in it must converge to a non-negative constant at infinity. These two behaviors cannot occur simultaneously unless the spinor has constant pointwise norm. This work may be seen as partial generalization of results obtained by Taubes[Tau17a], and Nagy and Oliveira [NO19] for the Kapustin-Witten equations.

We demonstrate a universal mechanism of a class of instabilities in infrared regions for massless Abelian p-form gauge theories with topological interactions, which we call generalized chiral instabilities. Such instabilities occur in the presence of initial electric fields for the p-form gauge fields. We show that the dynamically generated magnetic fields tend to decrease the initial electric fields and result in configurations with linking numbers, which can be characterized by non-invertible global symmetries. The so-called chiral plasma instability and instabilities of the axion electrodynamics and (4+1)-dimensional Maxwell-Chern-Simons theory in electric fields can be described by the generalized chiral instabilities in a unified manner. We also illustrate this mechanism in the (2+1)-dimensional Goldstone-Maxwell model in electric field.

The Invariant Risk Minimization (IRM) principle was first proposed by Arjovsky et al. [2019] to address the domain generalization problem by leveraging data heterogeneity from differing experimental conditions. Specifically, IRM seeks to find a data representation under which an optimal classifier remains invariant across all domains. Despite the conceptual appeal of IRM, the effectiveness of the originally proposed invariance penalty has recently been brought into question. In particular, there exists counterexamples for which that invariance penalty can be arbitrarily small for non-invariant data representations. We propose an alternative invariance penalty by revisiting the Gramian matrix of the data representation. We discuss the role of its eigenvalues in the relationship between the risk and the invariance penalty, and demonstrate that it is ill-conditioned for said counterexamples. The proposed approach is guaranteed to recover an invariant representation for linear settings under mild non-degeneracy conditions. Its effectiveness is substantiated by experiments on DomainBed and InvarianceUnitTest, two extensive test beds for domain generalization.

We introduce a generalized attention mechanism for spherical domains, enabling Transformer architectures to natively process data defined on the two-dimensional sphere - a critical need in fields such as atmospheric physics, cosmology, and robotics, where preserving spherical symmetries and topology is essential for physical accuracy. By integrating numerical quadrature weights into the attention mechanism, we obtain a geometrically faithful spherical attention that is approximately rotationally equivariant, providing strong inductive biases and leading to better performance than Cartesian approaches. To further enhance both scalability and model performance, we propose neighborhood attention on the sphere, which confines interactions to geodesic neighborhoods. This approach reduces computational complexity and introduces the additional inductive bias for locality, while retaining the symmetry properties of our method. We provide optimized CUDA kernels and memory-efficient implementations to ensure practical applicability. The method is validated on three diverse tasks: simulating shallow water equations on the rotating sphere, spherical image segmentation, and spherical depth estimation. Across all tasks, our spherical Transformers consistently outperform their planar counterparts, highlighting the advantage of geometric priors for learning on spherical domains.

We study the problem of designing models for machine learning tasks defined on sets. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics poczos13aistats, to anomaly detection in piezometer data of embankment dams Jung15Exploration, to cosmology Ntampaka16Dynamical,Ravanbakhsh16ICML1. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

The composition gamma circ eta of Jordan curves gamma and eta in universal Teichm\"uller space is defined through the composition h_gamma circ h_eta of their conformal weldings. We show that whenever gamma and eta are curves of finite Loewner energy I^L, the energy of the composition satisfies $I^L(gamma circ eta) lesssim_K I^L(gamma) + I^L(eta), with an explicit constant in terms of the quasiconformal K of \gamma and \eta. We also study the asymptotic growth rate of the Loewner energy under n self-compositions \gamma^n := \gamma \circ \cdots \circ \gamma, showing limsup_{n rightarrow infty} 1{n}log I^L(gamma^n) lesssim_K 1, again with explicit constant. Our approach is to define a new conformally-covariant rooted welding functional W_h(y), and show W_h(y) \asymp_K I^L(\gamma) when h is a welding of \gamma and y is any root (a point in the domain of h). In the course of our arguments we also give several new expressions for the Loewner energy, including generalized formulas in terms of the Riemann maps f and g for \gamma which hold irrespective of the placement of \gamma on the Riemann sphere, the normalization of f and g, and what disks D, D^c \subset \mathbb{C} serve as domains. An additional corollary is that I^L(\gamma) is bounded above by a constant only depending on the Weil--Petersson distance from \gamma$ to the circle.

Learning physical simulations has been an essential and central aspect of many recent research efforts in machine learning, particularly for Navier-Stokes-based fluid mechanics. Classic numerical solvers have traditionally been computationally expensive and challenging to use in inverse problems, whereas Neural solvers aim to address both concerns through machine learning. We propose a general formulation for continuous convolutions using separable basis functions as a superset of existing methods and evaluate a large set of basis functions in the context of (a) a compressible 1D SPH simulation, (b) a weakly compressible 2D SPH simulation, and (c) an incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries included in the basis functions are key aspects of stability and accuracy. Our broad evaluation shows that Fourier-based continuous convolutions outperform all other architectures regarding accuracy and generalization. Finally, using these Fourier-based networks, we show that prior inductive biases, such as window functions, are no longer necessary. An implementation of our approach, as well as complete datasets and solver implementations, is available at https://github.com/tum-pbs/SFBC.

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

This work considers gradient-based mesh optimization, where we iteratively optimize for a 3D surface mesh by representing it as the isosurface of a scalar field, an increasingly common paradigm in applications including photogrammetry, generative modeling, and inverse physics. Existing implementations adapt classic isosurface extraction algorithms like Marching Cubes or Dual Contouring; these techniques were designed to extract meshes from fixed, known fields, and in the optimization setting they lack the degrees of freedom to represent high-quality feature-preserving meshes, or suffer from numerical instabilities. We introduce FlexiCubes, an isosurface representation specifically designed for optimizing an unknown mesh with respect to geometric, visual, or even physical objectives. Our main insight is to introduce additional carefully-chosen parameters into the representation, which allow local flexible adjustments to the extracted mesh geometry and connectivity. These parameters are updated along with the underlying scalar field via automatic differentiation when optimizing for a downstream task. We base our extraction scheme on Dual Marching Cubes for improved topological properties, and present extensions to optionally generate tetrahedral and hierarchically-adaptive meshes. Extensive experiments validate FlexiCubes on both synthetic benchmarks and real-world applications, showing that it offers significant improvements in mesh quality and geometric fidelity.

Generating 3D images of complex objects conditionally from a few 2D views is a difficult synthesis problem, compounded by issues such as domain gap and geometric misalignment. For instance, a unified framework such as Generative Adversarial Networks cannot achieve this unless they explicitly define both a domain-invariant and geometric-invariant joint latent distribution, whereas Neural Radiance Fields are generally unable to handle both issues as they optimize at the pixel level. By contrast, we propose a simple and novel 2D to 3D synthesis approach based on conditional diffusion with vector-quantized codes. Operating in an information-rich code space enables high-resolution 3D synthesis via full-coverage attention across the views. Specifically, we generate the 3D codes (e.g. for CT images) conditional on previously generated 3D codes and the entire codebook of two 2D views (e.g. 2D X-rays). Qualitative and quantitative results demonstrate state-of-the-art performance over specialized methods across varied evaluation criteria, including fidelity metrics such as density, coverage, and distortion metrics for two complex volumetric imagery datasets from in real-world scenarios.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, are inherently unable to scale to high dimensions, or use approximations for limiting quantities that result in biased training objectives. Riemannian Flow Matching bypasses these limitations and offers several advantages over previous approaches: it is simulation-free on simple geometries, does not require divergence computation, and computes its target vector field in closed-form. The key ingredient behind RFM is the construction of a relatively simple premetric for defining target vector fields, which encompasses the existing Euclidean case. To extend to general geometries, we rely on the use of spectral decompositions to efficiently compute premetrics on the fly. Our method achieves state-of-the-art performance on many real-world non-Euclidean datasets, and we demonstrate tractable training on general geometries, including triangular meshes with highly non-trivial curvature and boundaries.

We construct a time-dependent, incompressible, and uniformly-in-time Lipschitz continuous velocity field on T^3 that produces exponential growth of the magnetic energy along a subsequence of times, for every positive value of the magnetic diffusivity. Because this growth is not uniform in time but occurs only along a diverging sequence of times, we refer to the resulting mechanism as a limsup fast dynamo. Our construction is based on suitably rescaled Arnold-Beltrami-Childress (ABC) flows, each supported on long time intervals. The analysis employs perturbation theory to establish continuity of the exponential growth rate with respect to both the initial data and the diffusivity parameter. This proves the weak form of the fast dynamo conjecture formulated by Childress and Gilbert on T^3, but the considerably more challenging version proposed by Arnold on T^3 remains an open problem.

Applications of machine learning techniques for materials modeling typically involve functions known to be equivariant or invariant to specific symmetries. While graph neural networks (GNNs) have proven successful in such tasks, they enforce symmetries via the model architecture, which often reduces their expressivity, scalability and comprehensibility. In this paper, we introduce (1) a flexible framework relying on stochastic frame-averaging (SFA) to make any model E(3)-equivariant or invariant through data transformations. (2) FAENet: a simple, fast and expressive GNN, optimized for SFA, that processes geometric information without any symmetrypreserving design constraints. We prove the validity of our method theoretically and empirically demonstrate its superior accuracy and computational scalability in materials modeling on the OC20 dataset (S2EF, IS2RE) as well as common molecular modeling tasks (QM9, QM7-X). A package implementation is available at https://faenet.readthedocs.io.

This paper introduces a novel surrogate modeling framework for aerodynamic applications based on Neural Fields. The proposed approach, MARIO (Modulated Aerodynamic Resolution Invariant Operator), addresses non parametric geometric variability through an efficient shape encoding mechanism and exploits the discretization-invariant nature of Neural Fields. It enables training on significantly downsampled meshes, while maintaining consistent accuracy during full-resolution inference. These properties allow for efficient modeling of diverse flow conditions, while reducing computational cost and memory requirements compared to traditional CFD solvers and existing surrogate methods. The framework is validated on two complementary datasets that reflect industrial constraints. First, the AirfRANS dataset consists in a two-dimensional airfoil benchmark with non-parametric shape variations. Performance evaluation of MARIO on this case demonstrates an order of magnitude improvement in prediction accuracy over existing methods across velocity, pressure, and turbulent viscosity fields, while accurately capturing boundary layer phenomena and aerodynamic coefficients. Second, the NASA Common Research Model features three-dimensional pressure distributions on a full aircraft surface mesh, with parametric control surface deflections. This configuration confirms MARIO's accuracy and scalability. Benchmarking against state-of-the-art methods demonstrates that Neural Field surrogates can provide rapid and accurate aerodynamic predictions under the computational and data limitations characteristic of industrial applications.

We study charged dilaton black branes in AdS_4. Our system involves a dilaton phi coupled to a Maxwell field F_{munu} with dilaton-dependent gauge coupling, {1over g^2} = f^2(phi). First, we find the solutions for extremal and near extremal branes through a combination of analytical and numerical techniques. The near horizon geometries in the simplest cases, where f(phi) = e^{alphaphi}, are Lifshitz-like, with a dynamical exponent z determined by alpha. The black hole thermodynamics varies in an interesting way with alpha, but in all cases the entropy is vanishing and the specific heat is positive for the near extremal solutions. We then compute conductivity in these backgrounds. We find that somewhat surprisingly, the AC conductivity vanishes like omega^2 at T=0 independent of alpha. We also explore the charged black brane physics of several other classes of gauge-coupling functions f(phi). In addition to possible applications in AdS/CMT, the extremal black branes are of interest from the point of view of the attractor mechanism. The near horizon geometries for these branes are universal, independent of the asymptotic values of the moduli, and describe generic classes of endpoints for attractor flows which are different from AdS_2times R^2.

Super-resolution (SR) advances astronomical imaging by enabling cost-effective high-resolution capture, crucial for detecting faraway celestial objects and precise structural analysis. However, existing datasets for astronomical SR (ASR) exhibit three critical limitations: flux inconsistency, object-crop setting, and insufficient data diversity, significantly impeding ASR development. We propose STAR, a large-scale astronomical SR dataset containing 54,738 flux-consistent star field image pairs covering wide celestial regions. These pairs combine Hubble Space Telescope high-resolution observations with physically faithful low-resolution counterparts generated through a flux-preserving data generation pipeline, enabling systematic development of field-level ASR models. To further empower the ASR community, STAR provides a novel Flux Error (FE) to evaluate SR models in physical view. Leveraging this benchmark, we propose a Flux-Invariant Super Resolution (FISR) model that could accurately infer the flux-consistent high-resolution images from input photometry, suppressing several SR state-of-the-art methods by 24.84% on a novel designed flux consistency metric, showing the priority of our method for astrophysics. Extensive experiments demonstrate the effectiveness of our proposed method and the value of our dataset. Code and models are available at https://github.com/GuoCheng12/STAR.

Anomalies in transverse Ward--Takahashi identities are studied, allowing discussion of the feasibility of anomalies arising in general non-symmetry Ward--Takahashi identities. We adopt the popular Fujikawa's method and rigorous dimensional renormalization to verify the existence of transverse anomalies to one-loop order and any loop order, respectively. The arbitrariness of coefficients of transverse anomalies is revealed, and a way out is also proposed after relating transverse anomalies to Schwinger terms and comparing symmetry and non-symmetry anomalies. Papers that claim the non-existence of transverse anomalies are reviewed to find anomalies hidden in their approaches. The role played by transverse anomalies is discussed.

In this paper, we consider the stability of the Lamb dipole solution of the two-dimensional Euler equations in R^{2} and question under which initial disturbance the Lamb dipole is stable, motivated by experimental work on the formation of a large vortex dipole in two-dimensional turbulence. We assume (O) odd symmetry for the x_2-variable and (N) non-negativity in the upper half plane for the initial disturbance of vorticity, and establish the stability theorem of the Lamb dipole without assuming (F) finite mass condition. The proof is based on a new variational characterization of the Lamb dipole using an improved energy inequality.

We introduce a classification conjecture for kappa-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of Z_2^2times O_3-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

This paper discusses a simple and explicit toy-model example of the categorical Hopfield equations introduced in previous work of Manin and the author. These describe dynamical assignments of resources to networks, where resources are objects in unital symmetric monoidal categories and assignments are realized by summing functors. The special case discussed here is based on computational resources (computational models of neurons) as objects in a category of DNNs, with a simple choice of the endofunctors defining the Hopfield equations that reproduce the usual updating of the weights in DNNs by gradient descent.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

Interpretability methods are valuable only if their explanations faithfully describe the explained model. In this work, we consider neural networks whose predictions are invariant under a specific symmetry group. This includes popular architectures, ranging from convolutional to graph neural networks. Any explanation that faithfully explains this type of model needs to be in agreement with this invariance property. We formalize this intuition through the notion of explanation invariance and equivariance by leveraging the formalism from geometric deep learning. Through this rigorous formalism, we derive (1) two metrics to measure the robustness of any interpretability method with respect to the model symmetry group; (2) theoretical robustness guarantees for some popular interpretability methods and (3) a systematic approach to increase the invariance of any interpretability method with respect to a symmetry group. By empirically measuring our metrics for explanations of models associated with various modalities and symmetry groups, we derive a set of 5 guidelines to allow users and developers of interpretability methods to produce robust explanations.

Within the context of a Chern-Simons running-vacuum-model (RVM) cosmology, one expects an early-matter dominated (eMD) reheating period after RVM inflation driven by the axion field. Treating thus in this work Chern-Simons RVM cosmology as an effective f(R) gravity theory characterized by logarithmic corrections of the spacetime curvature, we study the gravitational-wave (GW) signal induced by the nearly-scale invariant inflationary adiabatic curvature perturbations during the transition from the eMD era driven by the axion to the late radiation-dominated era. Remarkably, by accounting for the extra GW scalaron polarization present within f(R) gravity theories, we find regions in the parameter space of the theory where one is met with a distinctive induced GW signal with a universal f^6 high-frequency scaling compared to the f^7 scaling present in general relativity (GR). Interestingly enough, for axion masses m_a higher than 1 GeV and axion gauge couplings f_a above 10^{-3} Planck mass, one can produce induced GW spectra within the sensitivity bands of future GW observatories such as the Einstein Telescope (ET), the Laser Interferometer Space Antenna (LISA), the Big Bang Observer (BBO) and the Square Kilometer Arrays (SKA).

A sample of 60,410 bona fide optical quasars with astrometric proper motions in Gaia EDR3 and spectroscopic redshifts above 0.5 in an oval 8400 square degree area of the sky is constructed. Using orthogonal Zernike functions of polar coordinates, the proper motion fields are fitted in a weighted least-squares adjustment of the entire sample and of six equal bins of sorted redshifts. The overall fit with 37 Zernike functions reveals a statistically significant pattern, which is likely to be of instrumental origin. The main feature of this pattern is a chain of peaks and dips mostly in the R.A. component with an amplitude of 25~muas yr^{-1}. This field is subtracted from each of the six analogous fits for quasars grouped by redshifts covering the range 0.5 through 7.03, with median values 0.72, 1.00, 1.25, 1.52, 1.83, 2.34. The resulting residual patterns are noisier, with formal uncertainties up to 8~muas yr^{-1} in the central part of the area. We detect a single high-confidence Zernike term for R.A. proper motion components of quasars with redshifts around 1.52 representing a general gradient of 30 muas yr^{-1} over 150degr on the sky. We do not find any small- or medium-scale systemic variations of the residual proper motion field as functions of redshift above the 2.5,sigma significance level.

Gauging fermion parity and summing over spin structures are subtly distinct operations. We introduce 'bosonisation cohomology' groups H_B^{d+2}(X) to capture this difference, for theories in spacetime dimension d equipped with maps to some X. Non-trivial classes in H_B^{d+2}(X) contain theories for which (-1)^F is anomaly-free, but spin structure summation is anomalous. We formulate a sequence of cobordism groups whose failure to be exact is measured by H_B^{d+2}(X), and from here we compute it for X=pt. The result is non-trivial only in dimensions din 4Z+2, being due to the presence of gravitational anomalies. The first few are H_B^4=Z_2, probed by a theory of 8 Majorana-Weyl fermions in d=2, then H_B^8=Z_8, H_B^{12}=Z_{16}times Z_2. We rigorously derive a general formula extending this to every spacetime dimension. Along the way, we compile many general facts about (fermionic and bosonic) anomaly polynomials, and about spin and pin^- (co)bordism generators, that we hope might serve as a useful reference for physicists working with these objects. We briefly discuss some physics applications, including how the H_B^{12} class is trivialised in supergravity. Despite the name, and notation, we make no claim that H_B^bullet(X) actually defines a cohomology theory (in the Eilenberg-Steenrod sense).

Our aim is to contribute to quantum field theory (QFT) formalisms useful for descriptions of short time phenomena, dominant especially in heavy ion collisions. We formulate out-of-equilibrium QFT within the finite-time-path formalism (FTP) and renormalization theory (RT). The potential conflict of FTP and RT is investigated in g phi^3 QFT, by using the retarded/advanced (R/A) basis of Green functions and dimensional renormalization (DR). For example, vertices immediately after (in time) divergent self-energy loops do not conserve energy, as integrals diverge. We "repair" them, while keeping d<4, to obtain energy conservation at those vertices. Already in the S-matrix theory, the renormalized, finite part of Feynman self-energy Sigma_{F}(p_0) does not vanish when |p_0|rightarrowinfty and cannot be split to retarded and advanced parts. In the Glaser--Epstein approach, the causality is repaired in the composite object G_F(p_0)Sigma_{F}(p_0). In the FTP approach, after repairing the vertices, the corresponding composite objects are G_R(p_0)Sigma_{R}(p_0) and Sigma_{A}(p_0)G_A(p_0). In the limit drightarrow 4, one obtains causal QFT. The tadpole contribution splits into diverging and finite parts. The diverging, constant component is eliminated by the renormalization condition langle 0|phi|0rangle =0 of the S-matrix theory. The finite, oscillating energy-nonconserving tadpole contributions vanish in the limit trightarrow infty .

An influential conjecture by Witten states that there is an instanton Floer homology of four-manifolds with corners that in certain situations is isomorphic to Khovanov homology of a given knot K. The Floer chain complex is generated by Nahm pole solutions of the Kapustin-Witten equations on R^3 times R^+_y with an additional monopole-like singular behaviour along the knot K inside the three-dimensional boundary at y=0. The Floer differential is given by counting solutions of the Haydys-Witten equations that interpolate between Kapustin-Witten solutions along an additional flow direction R_s. This article investigates solutions of a decoupled version of the Kapustin-Witten and Haydys-Witten equations on R_s times R^3 times R^+_y, which in contrast to the full equations exhibit a Hermitian Yang-Mills structure and can be viewed as a lift of the extended Bogomolny equations (EBE) from three to five dimensions. Inspired by Gaiotto-Witten's approach of adiabatically braiding EBE-solutions to obtain generators of the Floer homology, we propose that there is an equivalence between adiabatic solutions of the decoupled Haydys-Witten equations and non-vertical paths in the moduli space of EBE-solutions fibered over the space of monopole positions. Moreover, we argue that the Grothendieck-Springer resolution of the Lie algebra of the gauge group provides a finite-dimensional model of this moduli space of monopole solutions. These considerations suggest an intriguing similarity between Haydys-Witten instanton Floer homology and symplectic Khovanov homology and provide a novel approach towards a proof of Witten's gauge-theoretic interpretations of Khovanov homology.

We delve into the physics-informed neural reconstruction of smoke and obstacles through sparse-view RGB videos, tackling challenges arising from limited observation of complex dynamics. Existing physics-informed neural networks often emphasize short-term physics constraints, leaving the proper preservation of long-term conservation less explored. We introduce Neural Characteristic Trajectory Fields, a novel representation utilizing Eulerian neural fields to implicitly model Lagrangian fluid trajectories. This topology-free, auto-differentiable representation facilitates efficient flow map calculations between arbitrary frames as well as efficient velocity extraction via auto-differentiation. Consequently, it enables end-to-end supervision covering long-term conservation and short-term physics priors. Building on the representation, we propose physics-informed trajectory learning and integration into NeRF-based scene reconstruction. We enable advanced obstacle handling through self-supervised scene decomposition and seamless integrated boundary constraints. Our results showcase the ability to overcome challenges like occlusion uncertainty, density-color ambiguity, and static-dynamic entanglements. Code and sample tests are at https://github.com/19reborn/PICT_smoke.

Recent approaches to arbitrary-scale single image super-resolution (ASR) use neural fields to represent continuous signals that can be sampled at arbitrary resolutions. However, point-wise queries of neural fields do not naturally match the point spread function (PSF) of pixels, which may cause aliasing in the super-resolved image. Existing methods attempt to mitigate this by approximating an integral version of the field at each scaling factor, compromising both fidelity and generalization. In this work, we introduce neural heat fields, a novel neural field formulation that inherently models a physically exact PSF. Our formulation enables analytically correct anti-aliasing at any desired output resolution, and -- unlike supersampling -- at no additional cost. Building on this foundation, we propose Thera, an end-to-end ASR method that substantially outperforms existing approaches, while being more parameter-efficient and offering strong theoretical guarantees. The project page is at https://therasr.github.io.

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

The observed cosmological constant may originate as the minimum value U_{min} of a scalar field potential, where the scalar field is frozen due to a large mass. If this vacuum is metastable, it may decay to a true vacuum either at present or in the future. Assuming its decay rate Gamma is comparable to the Hubble expansion rate H_0, we estimate the scale of true vacuum bubbles and analyze their evolution. We find that their initial formation scale is sub-millimeter and their tension causes rapid collapse if m gtrsim 1.7 cdot 10^{-3}, eV. For smaller masses, the bubbles expand at the speed of light. We extend our analysis to scalar-tensor theories with non-minimal coupling, finding that the nucleation scale of gravitational constant bubbles remains consistent with the sub-millimeter regime of General Relativity. The critical mass scale remains around 10^{-3},eV. A theoretical estimate at redshift z_{obs} sim 0.01 suggests an observable bubble radius of sim 50 Mpc, implying a gravitational transition triggered sim 300 Myr ago, with a present-day size approaching 100 Mpc. Additionally, we explore mass ranges (m < 10^{-3},eV) and non-minimal coupling xi ranges (10^{-8},eV^{2-n} - 10^{-1},eV^{2-n}) that lead to a variation Delta G/G_N within the 1%-7% range. We assume non-minimal coupling of the form F(phi)=1/kappa - xi phi^n, with kappa=8pi G_N and 2 leq n leq 9. Finally, we review various local physics or/and transition based proposed solutions to the Hubble tension, including ultra-late-time transitional models (z sim 0.01), screened fifth-force mechanisms, and the Lambda_{rm s}CDM model, which features a transition at z sim 2. We discuss observational hints supporting these scenarios and the theoretical challenges they face.

Small sample sizes are common in many disciplines, which necessitates pooling roughly similar datasets across multiple institutions to study weak but relevant associations between images and disease outcomes. Such data often manifest shift/imbalance in covariates (i.e., secondary non-imaging data). Controlling for such nuisance variables is common within standard statistical analysis, but the ideas do not directly apply to overparameterized models. Consequently, recent work has shown how strategies from invariant representation learning provides a meaningful starting point, but the current repertoire of methods is limited to accounting for shifts/imbalances in just a couple of covariates at a time. In this paper, we show how viewing this problem from the perspective of Category theory provides a simple and effective solution that completely avoids elaborate multi-stage training pipelines that would otherwise be needed. We show the effectiveness of this approach via extensive experiments on real datasets. Further, we discuss how this style of formulation offers a unified perspective on at least 5+ distinct problem settings, from self-supervised learning to matching problems in 3D reconstruction.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

Motivated by a variant of Atiyah-Floer conjecture proposed in L2 and its potential generalizations, we study in this article and its sequel as a first step properties of moduli spaces of Seiberg-Witten equations on a 3-dimensional cobordism with cylindrical ends (CCE) \(Y\), perturbed by closed 2-forms of the form \(r*d\ff+w\), where \(r\geq 1\), where \(\ff\) is a harmonic Morse function with certain linear growth at the ends of \(Y\), and \(w\) is a certain closed 2-form.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Proper evaluations are crucial for better understanding, troubleshooting, interpreting model behaviors and further improving model performance. While using scalar-based error metrics provides a fast way to overview model performance, they are often too abstract to display certain weak spots and lack information regarding important model properties, such as robustness. This not only hinders machine learning models from being more interpretable and gaining trust, but also can be misleading to both model developers and users. Additionally, conventional evaluation procedures often leave researchers unclear about where and how model fails, which complicates model comparisons and further developments. To address these issues, we propose a novel evaluation workflow, named Non-Equivariance Revealed on Orbits (NERO) Evaluation. The goal of NERO evaluation is to turn focus from traditional scalar-based metrics onto evaluating and visualizing models equivariance, closely capturing model robustness, as well as to allow researchers quickly investigating interesting or unexpected model behaviors. NERO evaluation is consist of a task-agnostic interactive interface and a set of visualizations, called NERO plots, which reveals the equivariance property of the model. Case studies on how NERO evaluation can be applied to multiple research areas, including 2D digit recognition, object detection, particle image velocimetry (PIV), and 3D point cloud classification, demonstrate that NERO evaluation can quickly illustrate different model equivariance, and effectively explain model behaviors through interactive visualizations of the model outputs. In addition, we propose consensus, an alternative to ground truths, to be used in NERO evaluation so that model equivariance can still be evaluated with new, unlabeled datasets.

Diffusion models based on permutation-equivariant networks can learn permutation-invariant distributions for graph data. However, in comparison to their non-invariant counterparts, we have found that these invariant models encounter greater learning challenges since 1) their effective target distributions exhibit more modes; 2) their optimal one-step denoising scores are the score functions of Gaussian mixtures with more components. Motivated by this analysis, we propose a non-invariant diffusion model, called SwinGNN, which employs an efficient edge-to-edge 2-WL message passing network and utilizes shifted window based self-attention inspired by SwinTransformers. Further, through systematic ablations, we identify several critical training and sampling techniques that significantly improve the sample quality of graph generation. At last, we introduce a simple post-processing trick, i.e., randomly permuting the generated graphs, which provably converts any graph generative model to a permutation-invariant one. Extensive experiments on synthetic and real-world protein and molecule datasets show that our SwinGNN achieves state-of-the-art performances. Our code is released at https://github.com/qiyan98/SwinGNN.

This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points. As this definition relies on points and not on tangent vectors, it can also be extended to geodesic spaces which are not Riemannian. For instance, in stratified spaces, it naturally allows principal subspaces that span several strata, which is impossible in previous generalizations of PCA. We show that barycentric subspaces locally define a submanifold of dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy of properly embedded linear subspaces of increasing dimension). We show that the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the subspaces of the flag (AUV). Barycentric subspaces are naturally nested, allowing the construction of hierarchically nested subspaces. Optimizing the AUV criterion to optimally approximate data points with flags of affine spans in Riemannian manifolds lead to a particularly appealing generalization of PCA on manifolds called Barycentric Subspaces Analysis (BSA).

We study superradiant scattering for a charged scalar field subject to Robin (mixed) boundary conditions on a charged BTZ black hole background. Scalar field modes having a real frequency do not exhibit superradiant scattering, independent of the boundary conditions applied. For scalar field modes with a complex frequency, no superradiant scattering occurs if the black hole is static. After exploring some regions of the parameter space, we provide evidence for the existence of superradiantly scattered modes with complex frequencies for a charged and rotating BTZ black hole. Most of the superradiantly scattered modes we find satisfy Robin (mixed) boundary conditions, but there are also superradiantly scattered modes with complex frequencies satisfying Dirichlet and Neumann boundary conditions. We explore the effect of the black hole and scalar field charge on the outgoing energy flux of these superradiantly scattered modes, and also investigate their stability.

In this paper, we adopt the Universal Manifold Embedding (UME) framework for the estimation of rigid transformations and extend it, so that it can accommodate scenarios involving partial overlap and differently sampled point clouds. UME is a methodology designed for mapping observations of the same object, related by rigid transformations, into a single low-dimensional linear subspace. This process yields a transformation-invariant representation of the observations, with its matrix form representation being covariant (i.e. equivariant) with the transformation. We extend the UME framework by introducing a UME-compatible feature extraction method augmented with a unique UME contrastive loss and a sampling equalizer. These components are integrated into a comprehensive and robust registration pipeline, named UMERegRobust. We propose the RotKITTI registration benchmark, specifically tailored to evaluate registration methods for scenarios involving large rotations. UMERegRobust achieves better than state-of-the-art performance on the KITTI benchmark, especially when strict precision of (1{\deg}, 10cm) is considered (with an average gain of +9%), and notably outperform SOTA methods on the RotKITTI benchmark (with +45% gain compared the most recent SOTA method).

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

The dynamical realisation of the equation of state p +rho =0 is studied. A non-pathological dynamics for the perturbations of such a system mimicking a dynamical cosmological constant (DCC) requires to go beyond the perfect fluid paradigm. It is shown that an anisotropic stress must be always present. The Hamiltonian of the system in isolation resembles the one of a Pais-Uhlenbeck oscillator and linear stability requires that it cannot be positive definite. The dynamics of linear cosmological perturbations in a DCC dominated Universe is studied in detail showing that when DCC is minimally coupled to gravity no dramatic instability is present. In contrast to what happens in a cosmological constant dominated Universe, the non-relativistic matter contrast is no longer constant and exhibits an oscillator behaviour at small scales while it grows weakly at large scales. In the gravitational waves sector, at small scales, the amplitude is still suppressed as the inverse power of the scale factor while it grows logarithmically at large scales. Also the vector modes propagate, though no growing mode is found.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Designing machine learning architectures for processing neural networks in their raw weight matrix form is a newly introduced research direction. Unfortunately, the unique symmetry structure of deep weight spaces makes this design very challenging. If successful, such architectures would be capable of performing a wide range of intriguing tasks, from adapting a pre-trained network to a new domain to editing objects represented as functions (INRs or NeRFs). As a first step towards this goal, we present here a novel network architecture for learning in deep weight spaces. It takes as input a concatenation of weights and biases of a pre-trained MLP and processes it using a composition of layers that are equivariant to the natural permutation symmetry of the MLP's weights: Changing the order of neurons in intermediate layers of the MLP does not affect the function it represents. We provide a full characterization of all affine equivariant and invariant layers for these symmetries and show how these layers can be implemented using three basic operations: pooling, broadcasting, and fully connected layers applied to the input in an appropriate manner. We demonstrate the effectiveness of our architecture and its advantages over natural baselines in a variety of learning tasks.

We study numerically the 6D (2,0) superconformal bootstrap using the soft-Actor-Critic (SAC) algorithm as a stochastic optimizer. We focus on the four-point functions of scalar superconformal primaries in the energy-momentum multiplet. Starting from the supergravity limit, we perform searches for adiabatically varied central charges and derive two curves for a collection of 80 CFT data (70 of these data correspond to unprotected long multiplets and 10 to protected short multiplets). We conjecture that the two curves capture the A- and D-series (2,0) theories. Our results are competitive when compared to the existing bounds coming from standard numerical bootstrap methods, and data obtained using the OPE inversion formula. With this paper we are also releasing our Python implementation of the SAC algorithm, BootSTOP. The paper discusses the main functionality features of this package.

We study the inflationary phenomenology of a rescaled Einstein-Gauss-Bonnet gravity. In this framework, the gravitational constant of the Einstein-Hilbert term is rescaled due to effective terms active in the high curvature era. Basically, the total theory is an F(R,G,phi) theory with the Gauss-Bonnet part contributing only a non-minimal coupling to the scalar field, so it is a theory with string theory origins and with a non-trivial F(R) gravity part. The F(R) gravity part in the high curvature regime contributes only a rescaled Einstein-Hilbert term and thus the resulting theory is effectively a rescaled version of a standard Einstein-Gauss-Bonnet theory. We develop the formalism of rescaled Einstein-Gauss-Bonnet gravity, taking in account the GW170817 constraints on the gravitational wave speed. We show explicitly how the rescaled theory affects directly the primordial scalar and tensor perturbations, and how the slow-roll and observational indices of inflation are affected by the rescaling of the theory. We perform a thorough phenomenological analysis of several models of interest and we show that is it possible to obtain viable inflationary theories compatible with the latest Planck data. Also among the studied models there are cases that yield a relatively large blue tilted tensor spectral index and we demonstrate that these models can lead to detectable primordial gravitational waves in the future gravitational wave experiments. Some of the scenarios examined, for specific values of the reheating temperature may be detectable by SKA, LISA, BBO, DECIGO and the Einstein Telescope.

We present a novel type of neural fields that uses general radial bases for signal representation. State-of-the-art neural fields typically rely on grid-based representations for storing local neural features and N-dimensional linear kernels for interpolating features at continuous query points. The spatial positions of their neural features are fixed on grid nodes and cannot well adapt to target signals. Our method instead builds upon general radial bases with flexible kernel position and shape, which have higher spatial adaptivity and can more closely fit target signals. To further improve the channel-wise capacity of radial basis functions, we propose to compose them with multi-frequency sinusoid functions. This technique extends a radial basis to multiple Fourier radial bases of different frequency bands without requiring extra parameters, facilitating the representation of details. Moreover, by marrying adaptive radial bases with grid-based ones, our hybrid combination inherits both adaptivity and interpolation smoothness. We carefully designed weighting schemes to let radial bases adapt to different types of signals effectively. Our experiments on 2D image and 3D signed distance field representation demonstrate the higher accuracy and compactness of our method than prior arts. When applied to neural radiance field reconstruction, our method achieves state-of-the-art rendering quality, with small model size and comparable training speed.

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

We provide a well-defined variational principle for 3-dimensional flat space Einstein gravity by adding one half of the Gibbons-Hawking-York boundary term to the bulk action. We check the 0-point function, recovering consistency with thermodynamics of flat space cosmologies. We then apply our result to calculate the 1-point functions in flat space Einstein gravity for the vacuum and all flat space cosmologies. The results are compatible with the ones for the zero mode charges obtained by canonical analysis.

Contours or closed planar curves are common in many domains. For example, they appear as object boundaries in computer vision, isolines in meteorology, and the orbits of rotating machinery. In many cases when learning from contour data, planar rotations of the input will result in correspondingly rotated outputs. It is therefore desirable that deep learning models be rotationally equivariant. In addition, contours are typically represented as an ordered sequence of edge points, where the choice of starting point is arbitrary. It is therefore also desirable for deep learning methods to be equivariant under cyclic shifts. We present RotaTouille, a deep learning framework for learning from contour data that achieves both rotation and cyclic shift equivariance through complex-valued circular convolution. We further introduce and characterize equivariant non-linearities, coarsening layers, and global pooling layers to obtain invariant representations for downstream tasks. Finally, we demonstrate the effectiveness of RotaTouille through experiments in shape classification, reconstruction, and contour regression.

In physics, Lagrangians provide a systematic way to describe laws governing physical systems. In the context of particle physics, they encode the interactions and behavior of the fundamental building blocks of our universe. By treating Lagrangians as complex, rule-based constructs similar to linguistic expressions, we trained a transformer model -- proven to be effective in natural language tasks -- to predict the Lagrangian corresponding to a given list of particles. We report on the transformer's performance in constructing Lagrangians respecting the Standard Model SU(3)times SU(2)times U(1) gauge symmetries. The resulting model is shown to achieve high accuracies (over 90\%) with Lagrangians up to six matter fields, with the capacity to generalize beyond the training distribution, albeit within architectural constraints. We show through an analysis of input embeddings that the model has internalized concepts such as group representations and conjugation operations as it learned to generate Lagrangians. We make the model and training datasets available to the community. An interactive demonstration can be found at: https://huggingface.co/spaces/JoseEliel/generate-lagrangians.