Efficient Sobolev approximation of linear parabolic PDEs in high dimensions

TL;DR

This paper demonstrates that a Monte Carlo Euler scheme can approximate solutions to high-dimensional linear parabolic PDEs in Sobolev norm without suffering from the curse of dimensionality, leveraging neural networks for efficient coefficient approximation.

Contribution

It introduces new error estimates for the Euler approximation of diffusions and their derivatives, enabling high-dimensional Sobolev norm approximation of PDE solutions.

Findings

Monte Carlo Euler scheme breaks the curse of dimensionality in Sobolev norm

Neural networks can efficiently approximate PDE coefficients in high dimensions

Error rates depend on discretization, simulations, and dimension

Abstract

In this paper, we study the error in first order Sobolev norm in the approximation of solutions to linear parabolic PDEs. We use a Monte Carlo Euler scheme obtained from combining the Feynman--Kac representation with a Euler discretization of the underlying stochastic process. We derive approximation rates depending on the time-discretization, the number of Monte Carlo simulations, and the dimension. In particular, we show that the Monte Carlo Euler scheme breaks the curse of dimensionality with respect to the first order Sobolev norm. Our argument is based on new estimates on the weak error of the Euler approximation of a diffusion process together with its derivative with respect to the initial condition. As a consequence, we obtain that neural networks are able to approximate solutions of linear parabolic PDEs in first order Sobolev norm without the curse of dimensionality if the coefficients of the PDEs admit an efficient approximation with neural networks.