mlsanityData to 7 Oct 2026

Robust discovery of partial differential equations in complex situations

arXiv 2106.00008 ↗First posted 31 May 2021cs.LG cs.AI math.OCcited by 21

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Abstract (arXiv)

Data-driven discovery of partial differential equations (PDEs) has achieved considerable development in recent years. Several aspects of problems have been resolved by sparse regression-based and neural network-based methods. However, the performances of existing methods lack stability when dealing with complex situations, including sparse data with high noise, high-order derivatives and shock waves, which bring obstacles to calculating derivatives accurately. Therefore, a robust PDE discovery framework, called the robust deep learning-genetic algorithm (R-DLGA), that incorporates the physics-informed neural network (PINN), is proposed in this work. In the framework, a preliminary result of potential terms provided by the deep learning-genetic algorithm is added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists to optimize the preliminary result and obtain the ultimately discovered PDE by eliminating the error compensation terms. The stability and accuracy of the proposed R-DLGA in several complex situations are examined for proof-and-concept, and the results prove that the proposed framework is able to calculate derivatives accurately with the optimization of PINN and possesses surprising robustness to complex situations, including sparse data with high noise, high-order derivatives, and shock waves.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 16 of its 30 references that are in the map. First posted on arXiv 31 May 2021.

Builds on

16 of 30 references.

References · 16

  1. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  2. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  3. 2020Weak SINDy For Partial Differential Equationscited by 154
  4. 2017Sparse identification of nonlinear dynamics for model predictive control in the low-data limitcited by 139
  5. 2017Sparse learning of stochastic dynamic equationscited by 76
  6. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 70
  7. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  8. 2020DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithmcited by 45
  9. 2018Data-driven discovery of PDEs in complex datasetscited by 42
  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. 2020Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Datacited by 19
4 more
  1. 2019Data-driven discovery of free-form governing differential equationscited by 16
  2. 2020Data-Driven Discovery of Coarse-Grained Equationscited by 14
  3. 2020Deep-learning based discovery of partial differential equations in integral form from sparse and noisy datacited by 14
  4. 2020A Lagrangian Dual-based Theory-guided Deep Neural Networkcited by 8

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 3; 2023: 5; 2024: 5; 2025: 3; 2026: 4.

21 papers.

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  1. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  2. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  3. 2022Integration of knowledge and data in machine learningcited by 13
  4. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  5. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 9
  6. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  7. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  8. 2023Discover governing differential equations from evolving systemscited by 5
  9. 2024An invariance constrained deep learning network for PDE discoverycited by 3
  10. 2024Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systemscited by 3
  11. 2024A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systemscited by 3
  12. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
9 more
  1. 2024Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regressioncited by 2
  2. 2025Evaluating PDE discovery methods for multiscale modeling of biological signalscited by 2
  3. 2021Sparsistent Model Discoverycited by 1
  4. 2026Joint discovery of governing partial differential equations from multi-source datasets by competitive optimizationcited by 1
  5. 2023Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systemscited by 0
  6. 2023Towards stable real-world equation discovery with assessing differentiating quality influencecited by 0
  7. 2024Automating the Discovery of Partial Differential Equations in Dynamical Systemscited by 0
  8. 2026Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Termscited by 0
  9. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0