A deep shotgun method for solving high-dimensional parabolic partial differential equations

TL;DR

This paper introduces a deep shotgun method for efficiently solving high-dimensional parabolic PDEs by utilizing data distribution instead of full trajectories, achieving high accuracy and performance in very high dimensions.

Contribution

The paper presents a novel deep shotgun approach that improves high-dimensional PDE solving by avoiding full trajectory simulation, enhancing efficiency and scalability.

Findings

Successfully applied to problems with up to 10,000 dimensions

Demonstrates superior performance over existing methods

Achieves high accuracy with reduced computational effort

Abstract

Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most existing methods requires simulating multiple trajectories of stochastic processes with a small step size of time discretization to ensure accuracy, hence having limited performance, especially when solving on a large time interval. To address such issue, we propose a deep "shotgun method" that does not exploit full trajectories, but only utilizes the data distribution of them. Numerical results including examples with dimensionality up to 10000 demonstrate the competitiveness of the proposed shotgun method in both performance and accuracy.