arXiv:2608. 09494v1 Announce Type: cross Abstract: In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing.
By Konrad Kleinberg, Thomas Kruse
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.
By Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See
arXiv:2508. 21571v2 Announce Type: replace Abstract: Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations.
By Bangti Jin, Longjun Wu
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:2604. 09361v3 Announce Type: replace Abstract: This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain.
By Zhangyong Liang, Tingfeng Wang, Xiaofei Zhao
arXiv:2607. 19167v1 Announce Type: cross Abstract: Motivated by the numerical computation of the Mean Escape Time (MET) $\tau:\Omega\to\mathbb{R}$ of a stochastic process from a bounded domain $\Omega\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $\rho$.
By Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou