[2402.07153] Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations

Mathematics > Numerical Analysis arXiv:2402.07153 (math) [Submitted on 11 Feb 2024 (v1), last revised 27 Mar 2026 (this version, v3)] Title:Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations Authors:Beatrice Lorenz, Aras Bacho, Gitta Kutyniok View a PDF of the paper titled Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations, by Beatrice Lorenz and 2 other authors View PDF HTML (experimental) Abstract:This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the $H^1([0,T];L^2(\Omega))$-norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments. Subjects: Numerical Analysis (math.NA); Artificial Intelligence (cs.AI) MSC classes: 35L05, 68T07, 65M15, 35G50, 35A35 Cite as: arXiv:2402.07153 [math.NA]   (or arXiv:2402.07153v3 [math.NA] for this version)   https://doi.org/10.48550/arXiv.2402.07153 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Aras Bacho [view email] [v1] Sun, 11 Feb 2024 10:50:20 UTC (1,393 KB) [v2] Wed, 6 Mar 2024 00:...