A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schrödinger Eigenvalue Problems
This provides a theoretical foundation for neural network methods in quantum mechanics, addressing the curse of dimensionality for high-dimensional eigenvalue problems.
The paper tackles the problem of computing the ground state of the Schrödinger operator in high dimensions using two-layer neural networks, proving that the generalization error converges at a rate independent of dimension under a spectral Barron space assumption.
This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension $d$, under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.