Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks

• DOI: 10.1016/j.cma.2024.117448
• Publication Date: 2024-10-18T02:45:26Z
• Authors: Zheyuan Hu; Kenji Kawaguchi; Zhongqiang Zhang; George Em Karniad...
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; I.2.7; FOS: Mathematics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); Dynamical Systems (math.DS); Mathematics - Dynamical Systems; F.2.2; F.2.2; I.2.7; Machine Learning (cs.LG)

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.<br/>15 pages<br/>