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At least 19 recordsLinked to original sources
arXiv2026

Learning Spectrally Optimised Mesh-Free Discretisations

Numerical methods for partial differential equations (PDEs) discretise differential operators, ideally reproducing the action of the continuous operators across all wavenumbers permitted by a given discretisation. Spectral-type methods approach this ideal, but rely on structured grids or high-order meshes that are hard to generate for complex geometries. In contrast, mesh-free methods are geometrically flexible, yet how faithfully they reproduce the operator across resolved scales is strongly influenced by a heuristically chosen kernel, which selects one of many weight sets satisfying the same consistency conditions without regard to the resulting spectral response. To address this, we introduce Spectrally optimised Neural Discretisations (SpeND), a framework that learns the map from local stencil geometry to discretisation weights on unstructured point clouds. A projection layer constrains every predicted stencil to the affine set defined by the discrete moment conditions, so polynomial consistency, and hence formal order of accuracy, holds exactly. Since consistency is enforced by the architecture, the weights can additionally be optimised for accuracy on a prescribed function space. Herein, we use Fourier modes and target the exact differentiation response over a chosen wavenumber band, so training is unsupervised. The loss function is a design interface: changing how it weights wavenumbers yields operators with distinct accuracy profiles. The trained discrete differential operators are PDE-agnostic and are applied without retraining to the Poisson, Burgers and Navier--Stokes equations, to near-boundary stencils, and across resolutions. At the same order and resolution, SpeND matches or improves on established mesh-free discretisations, and has been shown to reduce the wall-clock time between $3-20\times$ to achieve equivalent error as the baselines.

physics.comp-ph↗
arXiv2026

Observer-robust energy condition verification for warp drive spacetimes

Energy-condition tests for warp-drive spacetimes must account for all admissible causal directions at each point. We use the classical S-lemma to express the null, weak, and strong conditions as $4\times4$ linear matrix inequalities; the dominant condition requires two such tests. These criteria require neither Hawking-Ellis classification nor a rapidity cutoff. For the null condition in a fixed orthonormal tetrad, the optimized multiplier margin equals half the minimum normalized null-energy contraction. Interval evaluation of the metric and curvature provides pointwise certificates and global bounds on its minimum over the bubble wall. On flat unit-lapse slices, the momentum constraint relates Eulerian momentum to shift vorticity. For smooth shifts on all of Euclidean space with bounded vorticity, momentum vanishes identically precisely for a gradient plus rigid rotation. For shifts linear in speed, integrated negative Eulerian energy scales exactly quadratically when finite, with a fixed profile and integration domain on these slices. We implement the tests in Warpax, a JAX toolkit, and compare four warp-drive geometries at matched parameters across subluminal and superluminal speeds. At the reference parameters, the global bounds establish null-energy violation in all four bubble walls. The ideal irrotational Rodal profile has zero Eulerian momentum and is everywhere Type I; Type-IV regions are detected in the other sampled walls. For Rodal, the Eulerian reading misses about $73\%$ of the sampled wall weak-energy violations. These numerical type labels and sampled fractions are distinct from the interval bounds. Finite-segment null-geodesic integrals and flat-space quantum-inequality estimates supplement the pointwise analysis.

gr-qc↗
arXiv2026

Projected Neural Differential Equations for Learning Constrained Dynamics

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.

cs.LG↗
arXiv2026

Computing statistical Euler limits of the Navier--Stokes equations in three dimensions

We develop a Monte Carlo lattice Boltzmann method to compute statistical solutions to the three-dimensional incompressible Navier--Stokes and Euler equations. Entropic space-time adaptive relaxation of the higher order kinetic moments yields stable numerical solutions with decreasing viscosity. We provide a convergence analysis that is conditional on four explicitly stated assumptions regarding the discrete dynamics. Under diffusive scaling, the laws of the discrete ensemble converge along a subsequence to a limit satisfying the Foias--Temam Liouville formulation of the Navier--Stokes equations. Consequently, provided the structure function scaling holds uniformly, the vanishing viscosity limit of these measures satisfies the multi-point statistical Euler hierarchy of Fjordholm, Mishra, and Weber. The limit measures inherit the known weak-strong uniqueness principle on the interval of existence of a strong Euler solution. Under explicit scaling assumptions, a Kuznetsov-type argument yields a fractional 1-Wasserstein convergence rate. We present three-dimensional computations of time-dependent statistical solutions along the inviscid limit of the incompressible Navier--Stokes equations together with convergence measurements in the Wasserstein metric. Numerical experiments on a randomized Taylor--Green vortex with 24-dimensional initial uncertainty recover Kolmogorov's K41 scaling for energy spectra and structure functions, exhibit the failure of pathwise strong convergence, and yield Wasserstein convergence rates of about $0.5$ at the onset of turbulence. Finally, error measurements with respect to spectral hyperviscosity computations indicate that the computed limit measure is independent of the numerical regularization.

math.NA↗
arXiv2026

Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators

Machine learning surrogates based on neural operators have shown broad applicability in solving forward PDE problems. However, eigenvalue problems, in which an eigenparameter and one of several valid eigenmodes must be simultaneously solved, remain difficult because standard operator learning formulations assume a unique input-output map. This work demonstrates that Fourier Neural Operators (FNOs), combined with wavelet-based encodings of PDE inputs, can learn and predict multiple eigenmodes of the elastic wave equation, corresponding to deformation modes of acoustic waves propagating through arbitrary metamaterial geometries. We provide a mechanistic explanation and experimental evidence for why wavelet encodings are well matched to the dual spatial-spectral structure of the FNO, enabling deterministic mode selection on both continuous-valued and binary-valued geometries within a single model, and for why prediction accuracy varies with geometric discontinuities. For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU, while preserving high fidelity. These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.

cs.LG↗
arXiv2026

Picard Iteration for the Characteristic Initial Value Problem in Einstein Equations

We present an iteration algorithm for vacuum and Einstein scalar-field equations in double-null gauge, which transform the non-linear PDE into systems of ODE. The numerical realization combines characteristic constraint solves, LGL spectral elements, pole-free spherical operators, Galerkin projection, and independent first-order residual and consistency checks.

gr-qc↗
arXiv2026

Physical Law Ecology: mapping multi-mechanism ecologies as the zeroth step of data-driven scientific discovery

Every data-driven equation discovery method assumes (implicitly and without verification) that the target system obeys a single governing law ($K{=}1$). Here we show that this assumption is the primary bottleneck limiting scientific discovery in multi-mechanism systems, and introduce Physical Law Ecology, a framework that makes $K^*$ (the number of coexisting independent mechanisms) itself the first quantity to be determined from data. The framework automatically mines a pool of topologically distinct candidate equations, constructs a continuous dominance weight field across parameter space, and discovers analytic evolution laws governing mechanism succession---with optional monotonicity constraints encoding irreversible physics. Across four unrelated systems (elastomer mechanics, pool boiling, galactic dynamics, and droplet evaporation), BIC consistently identifies $K^*{=}3$ independent governing topologies. Applied to 163 SPARC galaxies (3,269 spatially resolved measurements), the framework autonomously recovers three gravitational laws whose coexistence provides evidence against the single-universal-acceleration hypothesis of MOND ($p<10^{-34}$). In engineering applications, multi-law weighted prediction reduces error by 67-72\% over single-equation baselines while retaining full interpretability. By establishing the determination of $K^*$ as the zeroth step of scientific discovery-prior to and independent of equation search---this work opens a direction orthogonal to existing symbolic regression: not finding better equations, but mapping the ecology of mechanisms that govern complex systems.

cs.SC↗
arXiv2026

Multi-Level-Set-Based Physics-Driven Neural Network to Solve 3-D Inverse Scattering Problems

This paper proposes a level-set-based physics-driven neural network solver (LSPDNN) for 3-D electromagnetic inverse scattering. To mitigate boundary blurring and reconstruction artifacts in voxel-wise contrast reconstruction, the proposed solver exploits the piecewise homogeneity of practical scatterers by representing unknown targets with multiple coordinate-dependent neural level-set components. Specifically, a soft-union multi-material model is proposed to separately describe the object support and material distribution. The global support is formed by the union of multiple level-set components, while the local contrast is determined by normalized component weights and learnable complex permittivity candidates. In addition, a model-consistent total variation (TV) regularization is imposed on the material-region indicators, rather than directly on the reconstructed contrast, to suppress fragmented material assignments without excessively smoothing material interfaces. An adaptive loss balancing strategy is further introduced to reduce the dependence on manually selected regularization weights. For each measurement instance, the neural level-set parameters and material candidates are optimized by minimizing a physics-consistent objective function. Numerical and experimental results demonstrate that LSPDNN can reconstruct scatterers with clear boundaries, more uniform material regions, and substantially reduced background artifacts. The results highlight the advantage of the neural level-set parameterization in challenging 3-D inverse scattering cases involving irregular shapes, closely spaced objects, multiple materials, and measurement noise.

cs.LG↗
arXiv2026

Effects of relational graph modularity and depth on the learning performance of neural networks

In recent years, graph-based machine learning techniques, such as reinforcement learning and graph neural networks, have garnered significant attention. While some recent studies have started to explore the relationship between the graph structure of neural networks and their predictive performance, they often limit themselves to a narrow range of model networks, particularly lacking mesoscale structures such as communities. Our work advances this area by conducting a more comprehensive investigation, incorporating realistic network structures characterized by heterogeneous degree distributions and community structures, which are typical characteristics of many real networks. These community structures offer a nuanced perspective on network architecture. Our analysis employs model networks such as random and scale-free networks, alongside a comparison with a biological neural network and its subsets for more detailed analysis. We examine the impact of these structural attributes on the performance of image classification tasks. Our findings reveal that structural properties do affect performance to some extent. Specifically, within moderate-depth architectures, networks featuring coherent, densely interconnected communities demonstrate enhanced learning capabilities. Crucially, we find that this advantage is strictly depth-dependent: extending the architecture to eight layers reverses the effect entirely. This comparison with the biological neural network emphasizes the relevance of our findings to real-world structures, suggesting an intriguing connection worth further exploration. This study contributes meaningfully to network science and machine learning, providing insights that could inspire the design of more biologically informed neural networks.

cs.LG↗
arXiv2026

Constitutive State-Space Modeling of Path-Dependent Plasticity: A Resolution-Consistent and Parallelizable Computational Framework

Data-driven constitutive models for path-dependent plasticity are commonly formulated using nonlinear recurrent neural networks, whose sequential state evolution limits parallel training and whose predictions may depend on the discretization of the applied strain path. We introduce a Constitutive State Space (CSS) model that reformulates structured state-space dynamics as an incremental constitutive operator. The strain increment is decomposed into magnitude and direction: the loading direction drives the latent state-space system, while the increment magnitude enters the zero-order-hold discretization of its continuous-time linear recurrence. This mechanics-tailored construction guarantees stationarity under zero increments, strongly reduces sensitivity to strain-path resolution, and retains the parallel-scan structure of S5 for efficient training on long constitutive histories. The CSS and Minimal State Cell (MSC) architectures are compared for four multiaxial path-dependent material models including isotropic J2 plasticity, pressure-sensitive foam plasticity, and combined isotropic-kinematic hardening. CSS matches or exceeds the prediction accuracy of the MSC, including one order of magnitude lower validation losses for the plastically incompressible materials. Importantly, CSS maintains low errors across large changes in strain-path discretization, whereas the MSC error increases substantially when evaluated at coarser resolutions than used for training. CSS trains substantially faster and requires fewer strain-stress pairs to attain comparable or better accuracy. Analysis of the learned state further reveals latent structure consistent with the dimensionality of the underlying physical constitutive models. These results establish mechanics-tailored structured state-space dynamics as a computational framework for efficient and discretization-robust data-driven constitutive modeling.

cs.LG↗
arXiv2026

Latent-MoE: Domain-Aware Mixture-of-Experts for PDEs with Multi-Regime Physics

Physics-informed neural networks (PINNs) struggle on PDEs whose governing physics varies across the domain. We trace this to a structural property of standard coordinate networks: their neural tangent kernel (NTK) is translation-variant and lets training points of large coordinate magnitude disproportionately influence predictions elsewhere, producing long-range coupling and gradient conflict during training. We show analytically and empirically that mixture-of-experts (MoE) architectures with centered, compact-support routers yield a uniformly banded NTK whose kernel-regression weights decay exponentially with distance, localizing the learning. Building on this, we propose \emph{Latent-MoE}, which interleaves domain-aware MoE blocks within a shared backbone. Unlike FB-PINNs or X-PINNs, which rigidly partition both the domain and the parameters so that the parameters on different subdomains are updated independently, Latent-MoE is designed to preserve the localization benefit of domain-aware routing while allowing capacity to flow across regions through the shared backbone. On standard homogeneous-physics benchmarks Latent-MoE is competitive with established baselines; on benchmarks with multi-stage time-variable physics, where global models and rigid domain decompositions both fall into spurious solutions, it improves over them by more than an order of magnitude, with markedly reduced gradient conflict during training.

cs.LG↗
arXiv2026

Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF-BPINN, a novel multi-fidelity framework that synergistically combines physics-informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low-fidelity simulations alongside sparse high-fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.

cs.LG↗
arXiv2026

High-Order Structure-Preserving SBP Finite Difference Methods for the Vlasov-Maxwell System on Matrix-Free GPUs

In this paper, we present a high-order, stable summation-by-parts (SBP) finite difference method for solving the Vlasov-Maxwell system in a 2D2V phase space. Central SBP operators for the advection terms are not stable when the solution becomes non-smooth and fine-scale filamentary structures develop, as is typical in high-dimensional Vlasov-Maxwell simulations. To address this issue, the method is stabilized using high-order upwind SBP operators. High-order explicit Runge-Kutta methods are employed for time integration. We prove that the fully discrete scheme exactly conserves mass and preserves momentum up to truncation error. Furthermore, we present a matrix-free implementation of the method on modern GPU architectures. A range of challenging benchmark problems is solved to demonstrate the accuracy, robustness, and performance of the proposed scheme.

math.NA↗
arXiv2026

A HIP-Compatible Accelerator Backend for Fourier-Bessel Particle-in-Cell Simulations on CPU/DCU Heterogeneous Clusters

FBPIC (Fourier-Bessel particle-in-cell) is a high-performance simulation code for relativistic plasma and accelerator physics. Its original accelerator backend relies on Numba CUDA, which limits its direct deployment on accelerators using the HIP (Heterogeneous-Compute Interface for Portability) programming environment, such as DCU (Deep Computing Unit) accelerators. In this work, we develop an accelerator backend compatible with HIP that enables FBPIC to run efficiently on DCU platforms while preserving its Python user interface and high level simulation workflow. For the evaluated LWFA (laser-wakefield acceleration) workloads, the proposed backend achieves 1.32-1.54x speedups over the original FBPIC implementation on an NVIDIA V100 GPU and enables efficient execution on the DCU platform. We also summarize the key lessons learned from porting FBPIC to the DCU platform. Multi-DCU experiments achieve a 1.88x strong-scaling speedup on four accelerators and a 2.72x increase in aggregate throughput at approximately 68\% weak-scaling efficiency, with communication analysis identifying inter-node communication and synchronization as the main scalability limitations. Beyond FBPIC, the proposed approach provides a practical reference for porting and optimizing other scientific computing applications developed with Python on heterogeneous accelerator platforms.

physics.comp-ph↗
arXiv2026

Euclidean Fourier Neural Operators

Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces as they are, by construction, independent of the grid resolution at which they are trained and evaluated. However, FNOs are not independent of the periodic domain they are applied to: their discrete spectral weights are indexed by integer Fourier mode numbers, which correspond to physical wavevectors. When applied to a different domain, the same trained weights act at different wavevectors, and the FNO silently represents a different operator. This makes FNOs unsuitable for tasks where transfer across domains is crucial. We propose Euclidean Fourier neural operators~(EFNOs) as a domain-independent alternative to FNOs. By parameterizing the spectral kernel as a continuous function of the physical wavevector, the EFNO can learn operators that act consistently across periodic domains of varying shape and size. We evaluate the EFNO on a simple heat equation and on a practically relevant materials science task of learning exchange-correlation potentials across different crystal structures, and demonstrate that the EFNO is able to generalize to unseen grid sizes and domains.

cs.LG↗
arXiv2026

A discontinuous Petrov-Galerkin finite-element framework for the simulation of microwave-heated flows

We present a high-order multiphysics solver for the simulation of microwave-heated flows. The solver couples a discontinuous Petrov-Galerkin (DPG) finite element method for the time-harmonic Maxwell equations with continuous Galerkin finite element methods for the heat equation and the incompressible Navier-Stokes equations. We validate the electromagnetic solver against multiple benchmark problems: wave propagation in a rectangular waveguide, a cavity problem with a singular solution, and a microwave-heated obstacle problem, comparing our results against numerical and experimental data from the literature. The results confirm the validity of the implementation and demonstrate its ability to perform adaptive mesh refinement using the DPG method's built-in error estimator. The final part of the study showcases the capabilities of the multiphysics framework through simulations of microwave-heated flow around obstacles with singular geometric features. These results highlight the potential of the proposed framework for the simulation and optimization of microwave-assisted chemical processes. Finally, the developed high-order multiphysics solver has a low memory footprint, since the electromagnetic solver relies on a Conjugate Gradient (CG) iterative solver and the fluid solver is implemented in a matrix-free fashion, making the overall approach scalable and well-suited for large-scale parallel simulations.

math.NA↗
arXiv2026

Deep Learning-Driven Peptide Classification in Biological Nanopores

Nanopore-based single-molecule sensing is a promising route to fast, low-cost disease diagnosis and protein sequencing: as an analyte such as a peptide or protein traverses a nanoscale pore, it modulates the ionic current, producing a resistive pulse whose signature is determined by the analyte's structure and its interactions with the pore. Translating these signatures into reliable molecular identities, however, is an open problem well suited for machine learning, as the signals are noisy, suffer from variations due to experimental conditions, and are difficult to featurize, which has so far limited classification accuracy. Here we translate the peptide identification problem into an image-classification task by transforming each resistive pulse into a scaleogram via the continuous wavelet transform, a representation that jointly encodes amplitude, frequency, and time in a form well suited for deep convolutional models. On a dataset of 42 peptides, recorded as six separate peptide ladders, this approach reaches a macro-averaged classification accuracy of $82\,\%$ on held-out events, an improvement of $8.6$ percentage points over the descriptor-based approach previously reported for the same dataset. We further show that the trained models tolerate substantial compression, retaining their accuracy with half of their weights set to zero and under 8-bit quantization, a prerequisite for deploying trained classifiers on embedded sensing hardware. Our results demonstrate how physically motivated signal representations can make complex single-molecule data tractable for modern learning algorithms, a step on the path towards point-of-care peptide and protein diagnostics.

cs.LG↗
arXiv2026

A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws

We propose a novel framework for deriving semi-discrete discontinuous-Galerkin (DG) methods using operator semigroups for scalar conservation laws, then apply it to construct a class of high-order OFDG-type schemes [13] satisfying infinitely many local entropy inequalities with general E-fluxes on non-uniform meshes. Such schemes are further generalized to systems of conservation laws in any number of space dimensions by using entropy stable numerical fluxes in the sense of [1]. Finally, we prove optimal error estimates for smooth solutions to nonlinear scalar conservation laws, and prove strong convergence for discontinuous solutions to strictly convex conservation laws via compensated compactness.

math.NA↗