Variational Monte Carlo Approach to Partial Differential Equations with Neural Networks

TL;DR

This paper introduces a variational Monte Carlo method using neural networks to solve high-dimensional partial differential equations, effectively modeling evolving probability densities with high accuracy on unbounded domains.

Contribution

It presents a novel variational approach that encodes the full probability density and adapts dynamically, enabling solutions in high dimensions and unbounded domains.

Findings

Excellent agreement with numerical solutions

Effective in high-dimensional regimes

Works on unbounded continuous domains

Abstract

The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here, we develop a variational approach for solving partial differential equations governing the evolution of high dimensional probability distributions. Our approach naturally works on the unbounded continuous domain and encodes the full probability density function through its variational parameters, which are adapted dynamically during the evolution to optimally reflect the dynamics of the density. For the considered benchmark cases we observe excellent agreement with numerical solutions as well as analytical solutions in regimes inaccessible to traditional computational approaches.