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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 partial differential equation with Neumann boundary condition. Using Barron’s representation of the solution IEEE Trans. Inform. Theory 39 (1993), pp. 930–945 with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the 2 2 -Wasserstein space of the set of parameter values defining the neural network. Inspired by the work from Bach and Chizat On the global convergence of gradient descent for over-parameterized models using optimal transport, 2018; ICM–International Congress of Mathematicians, EMS Press, Berlin, 2023, we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. Numerical experiments are given to show the potential of the method.
Dus et al. (Sat,) studied this question.