Error estimates of residual minimization using neural networks for linear PDEs
arXiv:2010.08019
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.
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- 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