Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes

TL;DR

This paper compares different discretization schemes for deep learning approaches to solve high-dimensional non-linear Kolmogorov equations, demonstrating potential accuracy improvements without increasing computational complexity.

Contribution

It introduces a comparison of various discretization schemes in deep learning methods for high-dimensional Kolmogorov equations, highlighting possible accuracy gains.

Findings

Certain discretization schemes improve accuracy

No increase in computational complexity with some schemes

Deep learning effectively solves high-dimensional PDEs

Abstract

Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black-Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced by E, Han, and Jentzen [1][2]. The main idea is to construct a deep network which is trained from the samples of discrete stochastic differential equations underlying Kolmogorov's equation. The network is able to approximate, numerically at least, the solutions of the Kolmogorov equation with polynomial complexity in whole spatial domains. In this contribution we study variants of the deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.