Adaptive deep density approximation for fractional Fokker-Planck equations

TL;DR

This paper introduces adaptive deep learning methods using flow-based models to efficiently solve fractional Fokker-Planck equations, overcoming challenges of traditional methods in high dimensions and unbounded domains.

Contribution

The paper develops novel adaptive deep density models, MCNF and GRBFNF, for stationary and time-dependent fractional Fokker-Planck equations, incorporating two fractional Laplacian approximation techniques.

Findings

Effective in high-dimensional, unbounded domains

Accurate approximation of fractional Laplacian

Adaptive training improves solution quality

Abstract

In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker-Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of the unbounded computation domain, a large number of dimensions and the nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, simplified KRnet, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models, MCNF and GRBFNF, to approximate stationary FPEs and MCTNF to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches.