# Discovering Symbolic Differential Equations with Symmetry Invariants

arXiv: https://arxiv.org/abs/2505.12083 · HTML: https://mlsanity.com/paper/2505.12083/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 67 references that are in the map.
- First posted on arXiv 17 May 2025.

## Builds on (most cited first, 30 of 44)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Symbolic Regression with a Learned Concept Library](https://mlsanity.com/paper/2409.09359/) (2024)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [Learning Symbolic Physics with Graph Networks](https://mlsanity.com/paper/1909.05862/) (2019)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [In-Context Symbolic Regression: Leveraging Large Language Models for Function Discovery](https://mlsanity.com/paper/2404.19094/) (2024)

## Built on by (most cited first, 6 of 6)

- [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [EqOD: Symmetry-Informed Stability Selection for PDE Identification](https://mlsanity.com/paper/2605.11524/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery](https://mlsanity.com/paper/2609.33663/) (2026)

## Abstract (arXiv)

Discovering symbolic differential equations from data uncovers fundamental dynamical laws underlying complex systems. However, existing methods often struggle with the vast search space of equations and may produce equations that violate known physical laws. In this work, we address these problems by introducing the concept of symmetry invariants in equation discovery. We leverage the fact that differential equations admitting a symmetry group can be expressed in terms of differential invariants of symmetry transformations. Thus, we propose to use these invariants as atomic entities in equation discovery, ensuring the discovered equations satisfy the specified symmetry. Our approach integrates seamlessly with existing equation discovery methods such as sparse regression and genetic programming, improving their accuracy and efficiency. We validate the proposed method through applications to various physical systems, such as fluid and reaction-diffusion, demonstrating its ability to recover parsimonious and interpretable equations that respect the laws of physics.