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

TL;DR

This work tackles the curse of dimensionality in fractional and tempered fractional PDEs by developing MC-tfPINN, extending MC-fPINN to tempered operators, and introducing a Gaussian-quadrature-based replacement for the 1D radial integrals. By using Gauss-Jacobi quadrature for radial components and Gauss-Laguerre quadrature in tempered cases, the approach reduces variance, hyperparameter sensitivity, and computational cost, enabling accurate solutions in very high dimensions up to $10^5$. The methods are validated on forward and inverse problems for both fractional and tempered fractional PDEs, demonstrating improved accuracy and substantially faster convergence compared to vanilla MC-based PINNs. The proposed framework offers a scalable, mesh-free tool for nonlocal PDEs with practical impact on high-dimensional modeling across physics, engineering, and related fields.

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.