# Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks Source: [Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks]() Collection index: [llms.txt]() Data updated: 2025-10-25T17:53:22.414Z Authors: Zheyuan Hu, Kenji Kawaguchi, Zhongqiang Zhang, George Em Karniadakis Published: 2024-06-17T16:26:18.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Solving Fractional Advection-Diffusion Equations on ADE 2D Case A2 (T=100)](): Solving time-space fractional Advection-Diffusion Equations; Time-space fractional ADE 2D case A2 T=100 (long-time simulation); Relative L2 Error: 0.006208. - [Solving time-space fractional Advection-Diffusion Equations on Time-space fractional ADE 3D, case B1, T=100 (long-time simulation)](): Solving time-space fractional Advection-Diffusion Equations; Time-space fractional ADE 3D, case B1, T=100 (long-time simulation); Relative L2 test error: 0.001222. - [Fractional Poisson Equation Solving on Unit Ball (d=3, test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(1+alpha/2)) 1.0 (test); Relative L2 Error (Test): 0.00118. - [Solving fractional Poisson equations on fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test); Relative L2 Test Error: 0.00605. - [Solving fractional Poisson equations on fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=3, u\_exact = (1 - \|\|x\|\|^2)^(alpha/2)) 1.0 (test); Relative L2 Test Error: 0.03. - [Solving fractional Poisson equations on fPE on Unit Ball (d=2, v1.0) (test)](): Solving fractional Poisson equations; fPE on Unit Ball (d=2, u\_exact = (1 - \|\|x\|\|^2)^(1+alpha/2)) 1.0 (test); Relative L2 Error: 0.00133. ## Original sources and implementation [Original paper]() [PDF]()