A machine learning solver for high-dimensional integrals: Solving Kolmogorov PDEs by stochastic weighted minimization and stochastic gradient descent through a high-order weak approximation scheme of SDEs with Malliavin weights

TL;DR

This paper presents a novel machine learning approach that efficiently solves high-dimensional Kolmogorov PDEs and SDE expectations using stochastic weighted minimization and gradient descent, avoiding the curse of dimensionality.

Contribution

It introduces a new method combining stochastic weighted minimization with high-order weak approximation schemes for SDEs, enabling accurate solutions in high dimensions.

Findings

Successfully solves PDEs and SDEs up to 100 dimensions

Uses second and third-order discretization schemes

Demonstrates effectiveness through numerical examples

Abstract

The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from the curse of dimensionality. Numerical examples for PDEs and SDEs up to 100 dimensions are shown by using second and third-order discretization schemes in order to demonstrate the effectiveness of our method.