# Finite Expression Methods for Discovering Physical Laws from Data

arXiv: https://arxiv.org/abs/2305.08342 · HTML: https://mlsanity.com/paper/2305.08342/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 49 references that are in the map.
- First posted on arXiv 15 May 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)
- [Identification of physical processes via combined data-driven and data-assimilation methods](https://mlsanity.com/paper/1810.11977/) (2018)
- [Modeling unknown dynamical systems with hidden parameters](https://mlsanity.com/paper/2202.01858/) (2022)

## Built on by (most cited first, 9 of 9)

- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics](https://mlsanity.com/paper/2409.09811/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformers](https://mlsanity.com/paper/2309.16816/) (2023)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Learning Epidemiological Dynamics via the Finite Expression Method](https://mlsanity.com/paper/2412.21049/) (2024)
- [H-FEX: A Symbolic Learning Method for Hamiltonian Systems](https://mlsanity.com/paper/2506.20607/) (2025)
- [Identifying Unknown Stochastic Dynamics via Finite expression methods](https://mlsanity.com/paper/2504.07085/) (2025)
- [AutoSR: Automatic Symbolic Regression by Searching Research States](https://mlsanity.com/paper/2608.16876/) (2026)

## Abstract (arXiv)

Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we shall present a novel deep symbolic learning method called the "finite expression method" (FEX) to discover governing equations within a function space containing a finite set of analytic expressions, based on observed dynamic data. The key concept is to employ FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions through convolutions. Our numerical results demonstrate that our FEX surpasses other existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across a range of problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Moreover, the results highlight FEX's flexibility and expressive power in accurately approximating symbolic governing equations.