# Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method

arXiv: https://arxiv.org/abs/2410.00835 · HTML: https://mlsanity.com/paper/2410.00835/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 1 Oct 2024.

## Builds on (most cited first, 19 of 19)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Finite Expression Methods for Discovering Physical Laws from Data](https://mlsanity.com/paper/2305.08342/) (2023)
- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [An unsupervised deep learning approach in solving partial integro-differential equations](https://mlsanity.com/paper/2006.15012/) (2020)
- [Temporal Difference Learning for High-Dimensional PIDEs with Jumps](https://mlsanity.com/paper/2307.02766/) (2023)
- [A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks](https://mlsanity.com/paper/2105.03480/) (2021)

## Built on by (most cited first, 3 of 3)

- [From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs](https://mlsanity.com/paper/2503.09986/) (2025)
- [Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains](https://mlsanity.com/paper/2510.22497/) (2025)
- [The finite expression method for turbulent dynamics with high-order moment recovery](https://mlsanity.com/paper/2605.10687/) (2026)

## Abstract (arXiv)

In this paper, we introduce a new finite expression method (FEX) to solve high-dimensional partial integro-differential equations (PIDEs). This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon.