mlsanityData to 14 Oct 2026

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

arXiv 2406.11708 ↗First posted 17 Jun 2024math.NA cs.LG cs.NAcited by 5

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Abstract (arXiv)

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.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 16 of its 38 references that are in the map. First posted on arXiv 17 Jun 2024.

Builds on

16 of 38 references.

References · 16

  1. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  2. 2017Generalization in Deep Learningcited by 146
  3. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  4. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
  5. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  6. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
  7. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  8. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  9. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  10. 2024Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equationscited by 22
  11. 2022Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equationscited by 20
  12. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
4 more
  1. 2020Numerical methods for nonlocal and fractional modelscited by 12
  2. 2023deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equationscited by 7
  3. 2024GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domainscited by 6
  4. 2022MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo samplingcited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 3; 2026: 1.

5 papers.

  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  3. 2025Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Networkcited by 1
  4. 2025More Consistent Accuracy PINN via Alternating Easy-Hard Trainingcited by 0
  5. 2026Stochastic Dimension Zeroth-Order Estimator: Stable and Memory-Efficient Training of PINNscited by 0