Deep adaptive basis Galerkin method for high-dimensional evolution equations with oscillatory solutions

TL;DR

This paper introduces a deep adaptive basis Galerkin method combining spectral-Galerkin and neural networks to efficiently solve high-dimensional oscillatory evolution equations, with proven error bounds and superior numerical performance.

Contribution

The paper proposes a novel DABG method that integrates spectral-Galerkin for time and DNNs for space, providing theoretical error estimates and demonstrating improved results over existing methods.

Findings

Error bounds depend on basis functions and network width.

Method outperforms existing DNN approaches in numerical tests.

Convergence is achieved for Barron-type solutions.

Abstract

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method, which employs the spectral-Galerkin method for the time variable of oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posterior estimates of the solution error, which is bounded by the minimal loss function and the term $O(N^{-m})$, where $N$ is the number of basis functions and $m$ characterizes the regularity of the e'quation. We also show that if the true solution is a Barron-type function, the error bound converges to zero as $M=O(N^p)$ approaches to infinity, where $M$ is the width of the used networks, and $p$ is a positive constant. Numerical examples, including high-dimensional linear evolution equations and the nonlinear Allen-Cahn equation, are presented to demonstrate the performance of the proposed DABG method is better than that of existing DNNs.