In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform.
arXiv:2608. 08872v1 Announce Type: new Abstract: In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators.
arXiv:2402. 00152v5 Announce Type: replace Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question.
The article discusses how automatic differentiation (AD) enables neural networks to compute derivatives of governing equations to machine precision, forming the backbone of physics-informed machine learning. However, the author argues that numerical exactness does not guarantee physical fidelity, as AD lacks awareness of the physical structure a solution must satisfy. The paper highlights that many physical constraints—such as convection directionality, diffusion, dispersion, conservation, thermodynamic consistency, symmetry, symplectic structure, positivity, monotonicity, and boundedness—are overlooked by AD, suggesting a need to rethink how PDE-driven neural surrogates are built.
The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.
arXiv:2606. 04623v1 Announce Type: new Abstract: High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds.
The paper introduces beignet, a neural field architecture that replaces the random Fourier feature embedding in physics-informed neural networks with a trainable multi‑resolution Fourier feature pyramid. By using Fourier interpolation and spectral derivatives, beignet efficiently computes spatial derivatives and scales accuracy through the feature pyramid rather than the neural network size. Experiments show that beignet achieves more accurate PDE solutions with fewer parameters than existing PINN methods and can reach machine‑precision residuals on the inviscid Burgers blowup problem using the Adam optimizer.
arXiv:2609. 05263v1 Announce Type: cross Abstract: We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions.
arXiv:2606. 04623v2 Announce Type: replace Abstract: High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics that evolve on symplectic manifolds.
arXiv:2609. 16406v1 Announce Type: cross Abstract: Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years.
arXiv:2606. 08871v1 Announce Type: cross Abstract: The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces.
The paper explores neural network solvers for infinity and p‑Laplace problems, employing Physics‑Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets). It addresses computational challenges for large p values (2 to 1000) across 2D and 3D domains, showing advantages over traditional mesh‑based solvers, especially in three dimensions. The authors provide conditional convergence results for PINNs, a universal approximation theorem for DeepONet on the parametric p‑Poisson problem, and validate their methods with numerical experiments comparing performance to conventional approaches.
arXiv:2510. 19399v2 Announce Type: replace Abstract: Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs).