Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schr\"odinger Eigenvalue Problems

TL;DR

This paper introduces a machine learning approach for solving high-dimensional Schrödinger eigenvalue problems, achieving dimension-independent convergence rates and improved accuracy by using a novel trial function construction and spectral Barron space assumptions.

Contribution

It develops a boundary-condition-preserving trial function method and derives explicit generalization error bounds that avoid the curse of dimensionality for Schrödinger eigenvalue problems.

Findings

Achieves dimension-independent convergence rates.

Eliminates boundary penalty errors for better accuracy.

Extends error bounds to normalized penalty methods.

Abstract

We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schr\"odinger operator on a $d$-dimensional hypercube with Dirichlet boundary conditions. The cut-off function technique is employed to construct trial functions that precisely satisfy the homogeneous boundary conditions. This approach eliminates the error caused by the standard boundary penalty method, improves the overall accuracy of the method, as demonstrated by the typical numerical examples. Under the assumption that the eigenfunctions belong to a spectral Barron space, we derive an explicit convergence rate of the generalization error of the proposed method, which does not suffer from the curse of dimensionality. We verify the assumption by proving a new regularity shift result for the eigenfunctions when the potential function belongs to an appropriate spectral Barron space. Moreover, we extend the generalization error bound to the normalized penalty method, which is widely used in practice.