Regularity Analysis and Tensor Neural Network Methods for Quasiperiodic Elliptic Equations

Abstract

This paper investigates quasiperiodic elliptic equations using a well-established projection method, by which the original problem is reformulated as a degenerate periodic variational problem defined on a higher-dimensional torus. The main analytical difficulty lies in the degeneracy of the projected torus problem: the variational formulation is coercive in the projected Sobolev spaces induced by the projection matrix, but fails to be coercive with respect to the standard Sobolev norm. We establish the well-posedness and regularity estimates within these projected Sobolev spaces. However, such projected Sobolev regularity alone is insufficient for standard spectral approximation: we show that it may result in arbitrarily slow Fourier convergence. To overcome this limitation, under a Diophantine condition for the projection matrix, together with appropriate assumptions on the coefficients and source term, we derive improved Sobolev regularity of the solution in standard Sobolev spaces.This provides a systematic mechanism for justifying the Sobolev regularity required in Fourier spectral convergence analysis, rather than imposing it a priori. Consequently, the improved Sobolev regularity enables quantitative Fourier projection error estimates for quasiperiodic problems. Inspired by these analytical results, we propose an adaptive tensor neural network Galerkin method that is naturally tailored to this degenerate high-dimensional periodic problem. Thanks to its tensor-product structure, high-dimensional variational integrals can be fully decoupled into one-dimensional quadratures, avoiding Monte Carlo sampling and yielding accurate deterministic numerical results. Numerical experiments on several quasiperiodic elliptic problems demonstrate the accuracy and efficiency of the proposed method.