mlsanityData to 14 Oct 2026

Data-driven discovery of PDEs in complex datasets

arXiv 1808.10788 ↗First posted 31 Aug 2018stat.ML cs.LG cs.NAcited by 42

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Abstract (arXiv)

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution.

Key facts. Cited by 42 papers in the map (data through 14 Oct 2026). First posted on arXiv 31 Aug 2018.

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Citing papers per year: 2021: 20; 2022: 6; 2023: 7; 2024: 3; 2025: 3; 2026: 3.

42 papers.

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  1. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  2. 2021Robust discovery of partial differential equations in complex situationscited by 21
  3. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  4. 2023Learning in latent spaces improves the predictive accuracy of deep neural operatorscited by 19
  5. 2021Physics-informed Spline Learning for Nonlinear Dynamics Discoverycited by 18
  6. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  7. 2023Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Datacited by 16
  8. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  9. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
  10. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  11. 2021Physics-constrained Deep Learning for Robust Inverse ECG Modelingcited by 12
  12. 2022Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order modelscited by 10
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  1. 2021NeuralPDE: Modelling Dynamical Systems from Datacited by 9
  2. 2023DynaBench: A benchmark dataset for learning dynamical systems from low-resolution datacited by 8
  3. 2021Physics-informed generative neural network: an application to troposphere temperature predictioncited by 7
  4. 2022A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacousticcited by 7
  5. 2021Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networkscited by 5
  6. 2023Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernelscited by 5
  7. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  8. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4
  9. 2023Q-Flow: Generative Modeling for Differential Equations of Open Quantum Dynamics with Normalizing Flowscited by 4
  10. 2021Deep-Learning Discovers Macroscopic Governing Equations for Viscous Gravity Currents from Microscopic Simulation Datacited by 3
  11. 2021Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equationscited by 3
  12. 2021Predicting Shallow Water Dynamics using Echo-State Networks with Transfer Learningcited by 3
  13. 2024An invariance constrained deep learning network for PDE discoverycited by 3
  14. 2021Model discovery in the sparse sampling regimecited by 2
  15. 2021Physics-Coupled Spatio-Temporal Active Learning for Dynamical Systemscited by 2
  16. 2021Multigoal-oriented dual-weighted-residual error estimation using deep neural networkscited by 2
  17. 2024GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equationscited by 2
  18. 2024Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networkscited by 2
  19. 2025Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Datacited by 2
  20. 2021Improved Surrogate Modeling of Fluid Dynamics with Physics-Informed Neural Networkscited by 1
  21. 2021Fully differentiable model discoverycited by 1
  22. 2021ST-PCNN: Spatio-Temporal Physics-Coupled Neural Networks for Dynamics Forecastingcited by 1
  23. 2021A Neural Network Ensemble Approach to System Identificationcited by 1
  24. 2026Debiased neural operators for estimating functionalscited by 1
  25. 2021Paradigm Shift Through the Integration of Physical Methodology and Data Sciencecited by 0
  26. 2022Wave simulation in non-smooth media by PINN with quadratic neural network and PML conditioncited by 0
  27. 2023Addressing Discontinuous Root-Finding for Subsequent Differentiability in Machine Learning, Inverse Problems, and Controlcited by 0
  28. 2025GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamicscited by 0
  29. 2026Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Datacited by 0
  30. 2026EqOD: Symmetry-Informed Stability Selection for PDE Identificationcited by 0