Regularity Analysis and Tensor Neural Network Methods for Quasiperiodic Elliptic Equations

TL;DR

This paper introduces a new tensor neural network-based approach for solving quasiperiodic elliptic equations, supported by theoretical regularity analysis and demonstrated through numerical experiments.

Contribution

It develops a novel adaptive tensor neural network method with theoretical regularity estimates for quasiperiodic elliptic problems.

Findings

High accuracy in high-dimensional integration achieved.

Method outperforms traditional schemes in efficiency.

Numerical experiments confirm theoretical regularity results.

Abstract

In this paper, we propose a novel machine learning method based on an adaptive tensor neural network subspace for solving quasiperiodic elliptic problems. To this end, we first provide a theoretical analysis of the associated quasiperiodic and periodic function spaces and establish regularity estimates for the quasiperiodic elliptic problems. In particular, under the Diophantine condition, we derive a suitable condition on the source term to guarantee the regularity of the solution, which provides a theoretical basis for the design of numerical schemes. An efficient numerical method is then designed by combining the projection method with tensor neural networks. Leveraging the special structure of tensor neural networks, high-dimensional integration can be performed directly and with high accuracy, without relying on Monte Carlo methods. Finally, several numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.