This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr\"odinger 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.

翻译：本文分析了用于计算Schr\'odinger操作器地面状态的两层神经网络在以美元维度超立方体计算地面状态方面的普遍误差。 我们证明,在假定地面状态位于光谱 Barron 空间的前提下,一般误差的趋同率独立于维度 $d$。 我们通过证明光谱Barron 空间地面状态的新的规律性估计来核实这种假设。 后一种假设是通过基于 Krein- Rutman 理论的固定点参数来实现的。