Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks

TL;DR

This paper introduces an improved Monte Carlo physics-informed neural network method for solving high-dimensional fractional and tempered fractional PDEs, significantly enhancing accuracy and convergence speed by replacing Monte Carlo sampling with Gaussian quadrature.

Contribution

The paper extends MC-fPINN to tempered fractional PDEs and replaces Monte Carlo sampling with Gaussian quadrature to reduce variance and improve performance in high dimensions.

Findings

Outperforms original MC-fPINN/MC-tfPINN in accuracy

Achieves faster convergence in up to 100,000 dimensions

Effective for both forward and inverse fractional PDE problems

Abstract

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.