# Generative Discovery of Partial Differential Equations by Learning from Math Handbooks

arXiv: https://arxiv.org/abs/2505.05869 · HTML: https://mlsanity.com/paper/2505.05869/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 59 references that are in the map.
- First posted on arXiv 9 May 2025.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [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)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [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)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Small Data, Big Decisions: Model Selection in the Small-Data Regime](https://mlsanity.com/paper/2009.12583/) (2020)
- [Symbol: Generating Flexible Black-Box Optimizers through Symbolic Equation Learning](https://mlsanity.com/paper/2402.02355/) (2024)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming](https://mlsanity.com/paper/2211.03134/) (2022)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data](https://mlsanity.com/paper/2309.04699/) (2023)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [LLM4ED: Large Language Models for Automatic Equation Discovery](https://mlsanity.com/paper/2405.07761/) (2024)
- [Distilling Wikipedia mathematical knowledge into neural network models](https://mlsanity.com/paper/2104.05930/) (2021)
- [Sparse discovery of differential equations based on multi-fidelity Gaussian process](https://mlsanity.com/paper/2401.11825/) (2024)

## Built on by (most cited first, 12 of 12)

- [OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inference](https://mlsanity.com/paper/2602.01493/) (2026)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation](https://mlsanity.com/paper/2607.10546/) (2026)
- [Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Data](https://mlsanity.com/paper/2603.22380/) (2026)
- [Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms](https://mlsanity.com/paper/2604.18414/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations](https://mlsanity.com/paper/2607.29665/) (2026)
- [Large language models for partial differential equation workflows](https://mlsanity.com/paper/2608.03600/) (2026)
- [Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields](https://mlsanity.com/paper/2608.27475/) (2026)
- [See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs](https://mlsanity.com/paper/2608.27869/) (2026)

## Abstract (arXiv)

Data driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data driven techniques face the dilemma of balancing search space with optimization efficiency. This study introduces a knowledge guided approach that incorporates existing PDEs documented in a mathematical handbook to facilitate the discovery process. These PDEs are encoded as sentence like structures composed of operators and basic terms, and used to train a generative model, called EqGPT, which enables the generation of free form PDEs. A loop of generation evaluation optimization is constructed to autonomously identify the most suitable PDE. Experimental results demonstrate that this framework can recover a variety of PDE forms with high accuracy and computational efficiency, particularly in cases involving complex temporal derivatives or intricate spatial terms, which are often beyond the reach of conventional methods. The approach also exhibits generalizability to irregular spatial domains and higher dimensional settings. Notably, it succeeds in discovering a previously unreported PDE governing strongly nonlinear surface gravity waves propagating toward breaking, based on real world experimental data, highlighting its applicability to practical scenarios and its potential to support scientific discovery.