Score-fPINN: Fractional Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck-Levy Equations

TL;DR

This paper presents a novel fractional score-based PINN method to efficiently solve high-dimensional Fokker-Planck-Levy equations, overcoming the curse of dimensionality and numerical issues in modeling complex stochastic processes.

Contribution

It introduces a fractional score function transformation and two methods, FSM and score-fPINN, to improve solving high-dimensional FPL equations with neural networks.

Findings

Demonstrates numerical stability in high dimensions

Effectively handles exponential decay in solutions

Outperforms traditional methods in complex stochastic modeling

Abstract

We introduce an innovative approach for solving high-dimensional Fokker-Planck-L\'evy (FPL) equations in modeling non-Brownian processes across disciplines such as physics, finance, and ecology. We utilize a fractional score function and Physical-informed neural networks (PINN) to lift the curse of dimensionality (CoD) and alleviate numerical overflow from exponentially decaying solutions with dimensions. The introduction of a fractional score function allows us to transform the FPL equation into a second-order partial differential equation without fractional Laplacian and thus can be readily solved with standard physics-informed neural networks (PINNs). We propose two methods to obtain a fractional score function: fractional score matching (FSM) and score-fPINN for fitting the fractional score function. While FSM is more cost-effective, it relies on known conditional distributions. On the other hand, score-fPINN is independent of specific stochastic differential equations (SDEs) but requires evaluating the PINN model's derivatives, which may be more costly. We conduct our experiments on various SDEs and demonstrate numerical stability and effectiveness of our method in dealing with high-dimensional problems, marking a significant advancement in addressing the CoD in FPL equations.