Solving time dependent Fokker-Planck equations via temporal normalizing flow

TL;DR

This paper introduces an adaptive, mesh-free machine learning method using temporal normalizing flows to efficiently solve high-dimensional, time-dependent Fokker-Planck equations without labeled data.

Contribution

It presents a novel approach that models solutions with temporal normalizing flows trained on the TFP loss, enabling high-dimensional, mesh-free solutions.

Findings

Effective in high-dimensional problems

Mesh-free and adaptable to various scenarios

No labeled data required for training

Abstract

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.