arXiv:2602. 19381v2 Announce Type: replace-cross Abstract: We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces.
By Ziang Chen, Liqiang Huang, Mengxuan Yang, Shengxuan Zhou
arXiv:2112. 08125v3 Announce Type: replace-cross Abstract: We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations.
By Carlo Marcati, Christoph Schwab
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. 16510v1 Announce Type: cross Abstract: This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters.
By Vijay Kumar, Gautam Singh
arXiv:2609.38916v1 Announce Type: new
Abstract: Any-dimensional machine learning models, such as graph neural networks (GNNs), can be naturally trained and evaluated on inputs of different sizes and...
By Wilson G. Gregory, George A. Kevrekidis, Ben Blum-Smith, Soledad Villar
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