# Finite Expression Method for Solving High-Dimensional Partial Differential Equations

arXiv: https://arxiv.org/abs/2206.10121 · HTML: https://mlsanity.com/paper/2206.10121/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 63 references that are in the map.
- First posted on arXiv 21 Jun 2022.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [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)
- [Bag of Tricks for Image Classification with Convolutional Neural Networks](https://mlsanity.com/paper/1812.01187/) (2018)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Reinforcement Learning for Combinatorial Optimization: A Survey](https://mlsanity.com/paper/2003.03600/) (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)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Neural Optimizer Search with Reinforcement Learning](https://mlsanity.com/paper/1709.07417/) (2017)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Benchmark Analysis of Representative Deep Neural Network Architectures](https://mlsanity.com/paper/1810.00736/) (2018)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Neural Network Approximation: Three Hidden Layers Are Enough](https://mlsanity.com/paper/2010.14075/) (2020)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Evolving Reinforcement Learning Algorithms](https://mlsanity.com/paper/2101.03958/) (2021)
- [Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons](https://mlsanity.com/paper/2107.02397/) (2021)
- [Elementary superexpressive activations](https://mlsanity.com/paper/2102.10911/) (2021)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2012.08023/) (2020)

## Built on by (most cited first, 21 of 21)

- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformers](https://mlsanity.com/paper/2309.16816/) (2023)
- [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)
- [Neural Control of Parametric Solutions for High-dimensional Evolution PDEs](https://mlsanity.com/paper/2302.00045/) (2023)
- [Temporal Difference Learning for High-Dimensional PIDEs with Jumps](https://mlsanity.com/paper/2307.02766/) (2023)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Closed-form Solutions: A New Perspective on Solving Differential Equations](https://mlsanity.com/paper/2405.14620/) (2024)
- [Learning Epidemiological Dynamics via the Finite Expression Method](https://mlsanity.com/paper/2412.21049/) (2024)
- [From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs](https://mlsanity.com/paper/2503.09986/) (2025)
- [On the Stochastic (Variance-Reduced) Proximal Gradient Method for Regularized Expected Reward Optimization](https://mlsanity.com/paper/2401.12508/) (2024)
- [Parsing the Language of Expression: Enhancing Symbolic Regression with Domain-Aware Symbolic Priors](https://mlsanity.com/paper/2503.09592/) (2025)
- [H-FEX: A Symbolic Learning Method for Hamiltonian Systems](https://mlsanity.com/paper/2506.20607/) (2025)
- [SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving](https://mlsanity.com/paper/2602.03816/) (2026)
- [Approximation of Solution Operators for High-dimensional PDEs](https://mlsanity.com/paper/2401.10385/) (2024)
- [Identifying Unknown Stochastic Dynamics via Finite expression methods](https://mlsanity.com/paper/2504.07085/) (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)
- [OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems](https://mlsanity.com/paper/2606.19145/) (2026)
- [Enhancing Transformer Representations of Symbolic ODE Expressions](https://mlsanity.com/paper/2609.24746/) (2026)

## Abstract (arXiv)

Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in designing numerical schemes that scale in dimension. This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution.