• Abstract

The paper contains approximation guarantees for neural networks that are trained with gradient flow, with error measured in the continuous $L_2(S^{d−1 })$-norm on the $d$-dimensional unit sphere and targets that are Sobolev smooth. The networks are fully connected of constant depth and increasing width. We show gradient flow convergence based on a neural tangent kernel (NTK) argument for the non-convex optimization of the second but last layer. Unlike standard NTK analysis, the continuous error norm implies an under-parametrized regime, possible by the natural smoothness assumption required for approximation. The typical over-parametrization re-enters the results in form of a loss in approximation rate relative to established approximation methods for Sobolev smooth functions.

• General Summary

• Abstract

This paper is devoted to the numerical resolution of McKean-Vlasov control problems via the class of mean-field neural networks introduced in our companion paper [Pham and Warin, Neural Netw., 168, 2023] in order to learn the solution on the Wasserstein space. We propose several algorithms either based on dynamic programming with control learning by policy or value iteration, or backward stochastic differential equation SDE from stochastic maximum principle with global or local loss functions. Extensive numerical results on different examples are presented to illustrate the accuracy of each of our eight algorithms. We discuss and compare the pros and cons of all the tested methods.

• General Summary