mlsanityData to 7 Oct 2026

Machine Learning for Partial Differential Equations

arXiv 2303.17078 ↗First posted 30 Mar 2023cs.LG cs.NA math.APcited by 20

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Abstract (arXiv)

Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being advanced by machine learning, including: 1) the discovery of new governing PDEs and coarse-grained approximations for complex natural and engineered systems, 2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and 3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 46 of its 141 references that are in the map. First posted on arXiv 30 Mar 2023.

Builds on

46 of 141 references.

References · 46

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  6. 2021Machine learning accelerated computational fluid dynamicscited by 461
  7. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  8. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  9. 2020Lagrangian Neural Networkscited by 388
  10. 2019Machine Learning for Fluid Mechanicscited by 380
  11. 2022Message Passing Neural PDE Solverscited by 368
  12. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
34 more
  1. 2020Discovering Symbolic Models from Deep Learning with Inductive Biasescited by 335
  2. 2021Choose a Transformer: Fourier or Galerkincited by 333
  3. 2017PDE-Net: Learning PDEs from Datacited by 304
  4. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  5. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  6. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  7. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  8. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 156
  9. 2020Weak SINDy For Partial Differential Equationscited by 154
  10. 2017Linearly-Recurrent Autoencoder Networks for Learning Dynamicscited by 142
  11. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  12. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  13. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 134
  14. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  15. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
  16. 2019Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systemscited by 96
  17. 2022Clifford Neural Layers for PDE Modelingcited by 91
  18. 2020Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphscited by 90
  19. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
  20. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 76
  21. 2021Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equationscited by 68
  22. 2017Time-lagged autoencoders: Deep learning of slow collective variables for molecular kineticscited by 64
  23. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  24. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  25. 2019Learning Symbolic Physics with Graph Networkscited by 52
  26. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  27. 2020Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear termscited by 32
  28. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  29. 2021Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)cited by 22
  30. 2019Deep Learning Models for Global Coordinate Transformations that Linearize PDEscited by 17
  31. 2020Enhancement of shock-capturing methods via machine learningcited by 16
  32. 2020Data-Driven Discovery of Coarse-Grained Equationscited by 14
  33. 2020FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equationscited by 14
  34. 2020Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Datacited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 10; 2025: 3; 2026: 3.

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  1. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  2. 2024Operator Learning: Algorithms and Analysiscited by 84
  3. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  4. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11
  5. 2023Pseudo-Hamiltonian neural networks for learning partial differential equationscited by 10
  6. 2024Neural SPH: Improved Neural Modeling of Lagrangian Fluid Dynamicscited by 10
  7. 2024Speeding up and reducing memory usage for scientific machine learning via mixed precisioncited by 8
  8. 2024Nonlinear model reduction for operator learningcited by 8
  9. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
  10. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  11. 2025SIMSHIFT: A Benchmark for Adapting Neural Surrogates to Distribution Shiftscited by 6
  12. 2024JAX-SPH: A Differentiable Smoothed Particle Hydrodynamics Frameworkcited by 3
8 more
  1. 2024Data-driven identification of latent port-Hamiltonian systemscited by 3
  2. 2025EquiNO: A Physics-Informed Neural Operator for Multiscale Simulationscited by 3
  3. 2024Gradient Flow Based Phase-Field Modeling Using Separable Neural Networkscited by 1
  4. 2025Towards scientific machine learning for granular material simulations -- challenges and opportunitiescited by 1
  5. 2026Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagatorscited by 1
  6. 2024Reducing Spatial Discretization Error on Coarse CFD Simulations Using an OpenFOAM-Embedded Deep Learning Frameworkcited by 0
  7. 2026From Data to Laws: Neural Discovery of Conservation Laws Without False Positivescited by 0
  8. 2026Paving the Way for Point Cloud Video Representation Learning Using A PDE Modelcited by 0