Error estimates of residual minimization using neural networks for linear PDEs

Abstract

We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations.

References in corpus (17)

• DGM: A deep learning algorithm for solving partial differential equations
• An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• fPINNs: Fractional Physics-Informed Neural Networks
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• A unified deep artificial neural network approach to partial differential equations in complex geometries
• Machine Learning of Linear Differential Equations using Gaussian Processes
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Understanding and mitigating gradient pathologies in physics-informed neural networks
• Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networks
• Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions
• D3M: A deep domain decomposition method for partial differential equations
• When and why PINNs fail to train: A neural tangent kernel perspective
• VarNet: Variational Neural Networks for the Solution of Partial Differential Equations
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs

Cited by in corpus (13)

• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Convergence Rate Analysis for Deep Ritz Method
• Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance
• Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model
• A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations
• Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
• Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
• A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs
• On the Representation of Solutions to Elliptic PDEs in Barron Spaces
• Notes on Exact Boundary Values in Residual Minimisation
• Analysis of Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Conditions