A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

DOI10.4208/cicp.OA-2021-0186zbMath1491.65118arXiv2109.01780OpenAlexW4220704858WikidataQ114021243 ScholiaQ114021243MaRDI QIDQ5077701

Dingwei Li, Yan-Ming Lai, Fengru Wang, Xiliang Lu, Yu Ling Jiao, Jerry Zhijian Yang, Yang Wang

Publication date: 19 May 2022

Published in: Communications in Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2109.01780

zbMATH Keywords

B-splinesRademacher complexityPINNs\(\mathrm{ReLU}^3\) neural network

Mathematics Subject Classification ID

Numerical computation using splines (65D07) Nonparametric estimation (62G05) Artificial neural networks and deep learning (68T07) Neural networks for/in biological studies, artificial life and related topics (92B20) Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12)

Related Items (6)

Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance ⋮ A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs ⋮ Learning Specialized Activation Functions for Physics-Informed Neural Networks ⋮ Improved Analysis of PINNs: Alleviate the CoD for Compositional Solutions ⋮ Convergence Analysis of the Deep Galerkin Method for Weak Solutions ⋮ A New Function Space from Barron Class and Application to Neural Network Approximation

Uses Software

• DeepXDE
• DGM
• FPINNs

Cites Work

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• Weak adversarial networks for high-dimensional partial differential equations
• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• Weak convergence and empirical processes. With applications to statistics
• DGM: a deep learning algorithm for solving partial differential equations
• nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications
• Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I
• Solving high-dimensional partial differential equations using deep learning
• Convergence Rate Analysis for Deep Ritz Method
• DeepXDE: A Deep Learning Library for Solving Differential Equations
• fPINNs: Fractional Physics-Informed Neural Networks

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