# DISCOVER: Deep identification of symbolically concise open-form PDEs via enhanced reinforcement-learning

arXiv: https://arxiv.org/abs/2210.02181 · HTML: https://mlsanity.com/paper/2210.02181/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 4 Oct 2022.

## Builds on (most cited first, 18 of 18)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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 learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [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)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [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)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problems](https://mlsanity.com/paper/2005.10756/) (2020)
- [Integration of knowledge and data in machine learning](https://mlsanity.com/paper/2202.10337/) (2022)
- [AutoKE: An automatic knowledge embedding framework for scientific machine learning](https://mlsanity.com/paper/2205.05390/) (2022)

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

- [LLM-Based Scientific Equation Discovery via Physics-Informed Token-Regularized Policy Optimization](https://mlsanity.com/paper/2602.10576/) (2026)
- [MOT-SR: Multi-Objective Tool-Augmented Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2607.29561/) (2026)
- [LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery](https://mlsanity.com/paper/2609.33663/) (2026)

## Abstract (arXiv)

The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a structure-aware recurrent neural network agent is designed and seamlessly combined with the sparse regression method to generate concise and open-form PDE expressions. All of the generated PDEs are evaluated by a meticulously designed reward function by balancing fitness to data and parsimony, and updated by the model-based reinforcement learning in an efficient way. Customized constraints and regulations are formulated to guarantee the rationality of PDEs in terms of physics and mathematics. The experiments demonstrate that our framework is capable of mining open-form governing equations of several dynamic systems, even with compound equation terms, fractional structure, and high-order derivatives, with excellent efficiency. Without the need for prior knowledge, this method shows great potential for knowledge discovery in more complicated circumstances with exceptional efficiency and scalability.