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.
By Ahmed Abdeljawad, Marcello Carioni, Elena Cordero
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.
By Yahong Yang, Juncai He
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.
By Ameya D. Jagtap
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.
By Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fern\'andez, Jun Liu
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.
By Liyi Feng, Yifa Tang, Yulin Xie, Ruili Zhang, Aiqing Zhu
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.
By Brandon Zhao, Yixuan Wang, Jonathan T. Barron, Katherine L. Bouman, Dor Verbin, Pratul P. Srinivasan