Approximation error analysis of some deep backward schemes for nonlinear PDEs

TL;DR

This paper analyzes the approximation errors of deep backward schemes for solving high-dimensional nonlinear PDEs, providing convergence rates and theoretical insights into neural network-based algorithms like MDBDP and DBDP.

Contribution

It offers the first detailed approximation error analysis and convergence rates for the MDBDP and DBDP schemes in neural network-based PDE solvers.

Findings

Convergence rate in terms of the number of neurons for linear PDEs.

Error bounds for semilinear PDEs using deep splitting schemes.

Numerical tests demonstrating the effectiveness of the proposed methods.

Abstract

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence properties. The methods rely on probabilistic representation of PDEs by backward stochastic differential equations (BSDEs) and their iterated time discretization. Our proposed algorithm, called deep backward multistep scheme (MDBDP), is a machine learning version of the LSMDP scheme of Gobet, Turkedjiev (Math. Comp. 85, 2016). It estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed by stochastic gradient descent. Our main theoretical contribution is to provide an approximation error analysis of the MDBDP scheme as well as the deep splitting (DS) scheme for semilinear PDEs designed in Beck, Becker, Cheridito, Jentzen, Neufeld (2019). We also supplement the error analysis of the DBDP scheme of Hur{\'e}, Pham, Warin (Math. Comp. 89, 2020). This yields notably convergence rate in terms of the number of neurons for a class of deep Lipschitz continuous GroupSort neural networks when the PDE is linear in the gradient of the solution for the MDBDP scheme, and in the semilinear case for the DBDP scheme. We illustrate our results with some numerical tests that are compared with some other machine learning algorithms in the literature.