Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations
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Abstract (arXiv)
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 H1([0,T];L2(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.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 14 of its 42 references that are in the map. First posted on arXiv 11 Feb 2024.
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References · 14
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- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
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- 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
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- 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
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- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
- 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
- 2021On the approximation of functions by tanh neural networkscited by 81
- 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
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