Adaptive Two-Layer ReLU Neural Network: II. Ritz Approximation to Elliptic PDEs

TL;DR

This paper introduces an adaptive neural network method for solving elliptic PDEs using Ritz approximation, providing effective initialization, error estimation, and demonstrating superior performance on complex diffusion problems.

Contribution

It develops an adaptive neuron enhancement approach with theoretical analysis and practical estimators for solving elliptic PDEs using Ritz approximation with neural networks.

Findings

Ritz approximation is the best in energy norm.

Numerical estimators effectively guide adaptive enhancement.

Method performs well on problems with interface singularities.

Abstract

In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical integration mesh such that the approximation accuracy is within the prescribed tolerance. Moreover, the ANE method provides a natural process for obtaining a good initialization which is crucial for training nonlinear optimization problem. The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.