A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations

TL;DR

This paper introduces a deep backward regression scheme for efficiently solving high-dimensional nonlinear PDEs, improving stability and accuracy over previous methods.

Contribution

It reformulates local backward losses via conditional expectations, reducing variance and enhancing numerical stability in high-dimensional PDE solutions.

Findings

DBR performs competitively on high-dimensional benchmarks.

It is more stable than previous DBDP methods.

Theoretical analysis shows half-order convergence under certain assumptions.

Abstract

We propose a deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations. Building on the DBDP method of Hur\'e, Pham, and Warin~\cite{HCPHWX20}, the proposed method reformulates the local backward losses through conditional expectations and trains the resulting regression problems sequentially in time. This conditional-expectation formulation replaces pathwise Brownian fluctuations in the Euler residual by their averaged effect and therefore provides an intrinsic variance-reduction mechanism before loss evaluation. In practice, the conditional expectations are approximated by local multi-path Monte Carlo averages, which leads to smoother training targets and improved numerical stability. Numerical experiments show that DBR performs competitively on standard high-dimensional benchmarks and is more stable than DBDP1 on the challenging unbounded benchmark considered in Example~2. Under an idealized population-loss minimization setting, we provide an error analysis and establish a half-order convergence result under suitable approximation and integrability assumptions. We also discuss an extension to variational inequalities.