标题： Two-layers neural networks for Schr{ö}dinger eigenvalue problems 显示英文标题

标题： Two-layers neural networks for Schr{ö}dinger eigenvalue problems

摘要： 本文的目的是分析使用两层神经网络在无限宽度下解决高维Schrödinger本征值问题的数值方案，该问题具有光滑的相互作用势和单位立方体上的Neumann边界条件，在任何维度中均适用。 更精确地说，与Schrödinger算子最低本征值相关的任何本征函数都是相关能量的单位L 2范数最小值。 利用Barron关于解的概率测度表示，该测度定义在参数值集合上，并遵循Bach和Chizat [1]最初提出的思路，通过在定义神经网络的参数值集合的2-Wasserstein空间上的约束梯度曲线动力学来最小化能量。 我们证明了这种约束梯度曲线的解的存在性。 此外，我们证明了如果它收敛，那么所表示的函数就是所考虑的Schrödinger算子的本征函数。 至少在我们所知的范围内，这是首次对这类分析进行研究以处理非凸泛函的最小化。 显示英文摘要

摘要： The aim of this article is to analyze numerical schemes using two-layer neural networks withinfinite width for the resolution of high-dimensional Schr{\"o}dinger eigenvalue problems with smoothinteraction potentials and Neumann boundary condition on the unit cube in any dimension. Moreprecisely, any eigenfunction associated to the lowest eigenvalue of the Schr{\"o}dinger operator is a unitL 2 norm minimizer of the associated energy. Using Barron's representation of the solution witha probability measure defined on the set of parameter values and following the approach initiallysuggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curvedynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Weprove the existence of solutions to this constrained gradient curve. Furthermore, we prove that,if it converges, the represented function is then an eigenfunction of the considered Schr{\"o}dingeroperator. At least up to our knowledge, this is the first work where this type of analysis is carriedout to deal with the minimization of non-convex functionals.