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Preprints, Working Papers, ... Year : 2023

Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks

Abstract

The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson-Neumann partial differential equations (PDEs) with Neumann boundary conditions. Using Barron's representation of the solution [1] with a measure of probability, the energy is minimized thanks to a gradient curve dynamic on the 2 Wasserstein space of parameters defining the neural network. Inspired by the work from Bach and Chizat [2, 3], we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. In contrast to the works [2, 3], the activation function we use here is not assumed to be homogeneous to obtain global convergence of the flow. Numerical experiments are given to show the potential of the method.
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Dates and versions

hal-04089961 , version 1 (05-05-2023)
hal-04089961 , version 2 (13-07-2023)

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  • HAL Id : hal-04089961 , version 2

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Mathias Dus, Virginie Ehrlacher. Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks. 2023. ⟨hal-04089961v2⟩
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