mlsanityData to 14 Oct 2026

Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)

arXiv 2106.11927 ↗First posted 9 Jun 2021cs.NE cs.LG physics.data-ancited by 41

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Abstract (arXiv)

Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which are essential for deepening our understanding of physical processes and predicting future responses. However, the PDEs of many real-world problems are uncertain, which calls for PDE discovery. We propose the symbolic genetic algorithm (SGA-PDE) to discover open-form PDEs directly from data without prior knowledge about the equation structure. SGA-PDE focuses on the representation and optimization of PDE. Firstly, SGA-PDE uses symbolic mathematics to realize the flexible representation of any given PDE, transforms a PDE into a forest, and converts each function term into a binary tree. Secondly, SGA-PDE adopts a specially designed genetic algorithm to efficiently optimize the binary trees by iteratively updating the tree topology and node attributes. The SGA-PDE is gradient-free, which is a desirable characteristic in PDE discovery since it is difficult to obtain the gradient between the PDE loss and the PDE structure. In the experiment, SGA-PDE not only successfully discovered nonlinear Burgers' equation, Korteweg-de Vries (KdV) equation, and Chafee-Infante equation, but also handled PDEs with fractional structure and compound functions that cannot be solved by conventional PDE discovery methods.

Key facts. Cited by 41 papers in the map (data through 14 Oct 2026). Builds on 12 of its 43 references that are in the map. First posted on arXiv 9 Jun 2021.

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12 of 43 references.

References · 12

  1. 2018DARTS: Differentiable Architecture Searchcited by 1,798
  2. 2016Neural Architecture Search with Reinforcement Learningcited by 1,499
  3. 2017Progressive Neural Architecture Searchcited by 487
  4. 2017PDE-Net: Learning PDEs from Datacited by 311
  5. 2019Deep Learning for Symbolic Mathematicscited by 270
  6. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  7. 2020Physics-informed learning of governing equations from scarce datacited by 167
  8. 2020Weak SINDy For Partial Differential Equationscited by 155
  9. 2020DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithmcited by 46
  10. 2019DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy datacited by 38
  11. 2019Deep Learning of Subsurface Flow via Theory-guided Neural Networkcited by 32
  12. 2020Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning methodcited by 29

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Citing papers per year: 2022: 7; 2023: 8; 2024: 6; 2025: 9; 2026: 11.

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  1. 2022PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Datacited by 20
  2. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  3. 2024LLM4ED: Large Language Models for Automatic Equation Discoverycited by 14
  4. 2022Integration of knowledge and data in machine learningcited by 13
  5. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  6. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 11
  7. 2023Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challengescited by 9
  8. 2026GENSR: Symbolic Regression Based in Equation Generative Spacecited by 9
  9. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  10. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  11. 2022D-CIPHER: Discovery of Closed-form Partial Differential Equationscited by 7
  12. 2022Learning Differential Operators for Interpretable Time Series Modelingcited by 7
29 more
  1. 2022AutoKE: An automatic knowledge embedding framework for scientific machine learningcited by 6
  2. 2023Discovering stochastic partial differential equations from limited data using variational Bayes inferencecited by 6
  3. 2023Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discoverycited by 6
  4. 2023Discover governing differential equations from evolving systemscited by 5
  5. 2025Adaptive parameters identification for nonlinear dynamics using deep permutation invariant networkscited by 5
  6. 2025Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamicscited by 4
  7. 2022DISCOVER: Deep identification of symbolically concise open-form PDEs via enhanced reinforcement-learningcited by 3
  8. 2024An invariance constrained deep learning network for PDE discoverycited by 3
  9. 2024Decoding complexity: how machine learning is redefining scientific discoverycited by 3
  10. 2024A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systemscited by 3
  11. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  12. 2024Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regressioncited by 2
  13. 2025Evaluating PDE discovery methods for multiscale modeling of biological signalscited by 2
  14. 2023Filtered Partial Differential Equations: a robust surrogate constraint in physics-informed deep learning frameworkcited by 1
  15. 2025CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equationscited by 1
  16. 2025Hybrid Adaptive Modeling in Process Monitoring: Leveraging Sequence Encoders and Physics-Informed Neural Networkscited by 1
  17. 2025An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulencecited by 1
  18. 2026Joint discovery of governing partial differential equations from multi-source datasets by competitive optimizationcited by 1
  19. 2026LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generationcited by 1
  20. 2026Efficient Constant Optimization for Symbolic Regression with GPU-Accelerated Tree-Based Genetic Programmingcited by 1
  21. 2023Towards true discovery of the differential equationscited by 0
  22. 2024Discovery and inversion of the viscoelastic wave equation in inhomogeneous mediacited by 0
  23. 2025Symbolic identification of tensor equations in multidimensional physical fieldscited by 0
  24. 2026EqOD: Symmetry-Informed Stability Selection for PDE Identificationcited by 0
  25. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0
  26. 2026See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEscited by 0
  27. 2026DASS: A Solver-Adaptable Dynamic Auxiliary Search Strategy for Symbolic Regressioncited by 0
  28. 2026Learning to Design SR Solvers: Harness Optimization for Symbolic Regressioncited by 0
  29. 2026Beyond Equation Search: Conjecture-Guided Discovery with SCOPE and ClosureBenchcited by 0