A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

Abstract

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

arXiv commentRevised the definition of Barron space and updated the proofs induced by the changes

References in corpus (14)

• DGM: A deep learning algorithm for solving partial differential equations
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Solving high-dimensional partial differential equations using deep learning
• Neural Tangent Kernel: Convergence and Generalization in Neural Networks
• A Mean Field View of the Landscape of Two-Layers Neural Networks
• On Lazy Training in Differentiable Programming
• A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks
• A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
• Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
• Poisson CNN: Convolutional neural networks for the solution of the Poisson equation on a Cartesian mesh
• Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
• Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't
• Error estimates of residual minimization using neural networks for linear PDEs
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs

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• On the Representation of Solutions to Elliptic PDEs in Barron Spaces
• Solving PDEs on Unknown Manifolds with Machine Learning
• Analysis of Deep Ritz Methods for Laplace Equations with Dirichlet Boundary Conditions
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