Generalization error estimates of a machine learning method for solving high-dimensional Schrödinger eigenvalue problems

We propose a machine learning method for computing the eigenvalues and eigenfunctions of the Schrödinger operator on a $d$-dimensional hypercube with Dirichlet boundary conditions. The eigenpairslie deep within the spectrum. The cut-off function technique is employed to construct trial functions that precisely satisfy the Dirichlet boundary condition. This approach outperforms the standard boundary penalty method, as demonstrated by numerical examples. Assuming that the eigenfunctions belong to a new spectral Barron space, we derive a dimension-free convergence rate $\mathcal{O}(n^{-1/4})$ for the generalization error bound, with all constants in the error bound being explicit and growing at most polynomially. This assumption is validated by proving a new regularity shift result for the eigenfunctions when the potential 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.