Deep Runge-Kutta schemes for BSDEs

TL;DR

This paper introduces a novel deep learning-based probabilistic scheme that combines high order Runge-Kutta methods with deep neural networks to efficiently solve high-dimensional semi-linear PDEs associated with BSDEs, improving accuracy and computational cost.

Contribution

It extends previous work by integrating high order Runge-Kutta schemes with deep learning for BSDEs, providing convergence analysis and demonstrating efficiency for schemes up to third order.

Findings

Crank-Nicolson scheme offers a good balance of accuracy and efficiency.

Higher order schemes (second and third order) improve precision.

Numerical results confirm the method's effectiveness in high-dimensional settings.

Abstract

We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear parabolic partial differential equations. Our approach notably extends the one introduced in [Hure Pham Warin 2020] for the implicit Euler scheme to schemes which are more efficient in terms of discrete-time error. We establish some convergence results for our implemented schemes under classical regularity assumptions. We also illustrate the efficiency of our method for different schemes of order one, two and three. Our numerical results indicate that the Crank-Nicolson schemes is a good compromise in terms of precision, computational cost and numerical implementation.