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

arXiv: https://arxiv.org/abs/2406.11708 · HTML: https://mlsanity.com/paper/2406.11708/

## 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 (most cited first, 16 of 16)

- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Generalization in Deep Learning](https://mlsanity.com/paper/1710.05468/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2402.07465/) (2024)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)
- [Numerical methods for nonlocal and fractional models](https://mlsanity.com/paper/2002.01401/) (2020)
- [deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations](https://mlsanity.com/paper/2309.07684/) (2023)
- [GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domains](https://mlsanity.com/paper/2405.00217/) (2024)
- [MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling](https://mlsanity.com/paper/2212.12984/) (2022)

## Built on by (most cited first, 5 of 5)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs](https://mlsanity.com/paper/2505.03595/) (2025)
- [Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network](https://mlsanity.com/paper/2504.01440/) (2025)
- [More Consistent Accuracy PINN via Alternating Easy-Hard Training](https://mlsanity.com/paper/2512.17607/) (2025)
- [Stochastic Dimension Zeroth-Order Estimator: Stable and Memory-Efficient Training of PINNs](https://mlsanity.com/paper/2603.24002/) (2026)

## 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.