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PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network

arXiv 1812.04426 ↗First posted 30 Nov 2018cs.LG cs.NA math.NAcited by 208

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Abstract (arXiv)

Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work citeLong2018PDE where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

Key facts. Cited by 208 papers in the map (data through 7 Oct 2026). First posted on arXiv 30 Nov 2018.

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Citing papers per year: 2021: 49; 2022: 43; 2023: 43; 2024: 22; 2025: 28; 2026: 23.

208 papers.

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  1. 2021Learning to Optimize: A Primer and A Benchmarkcited by 206
  2. 2021PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equationscited by 106
  3. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  4. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 64
  5. 2021Encoding physics to learn reaction-diffusion processescited by 61
  6. 2022Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree searchcited by 61
  7. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  8. 2022Signal Processing for Implicit Neural Representationscited by 46
  9. 2022Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamicscited by 44
  10. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  11. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 40
  12. 2021LEADS: Learning Dynamical Systems that Generalize Across Environmentscited by 39
38 more
  1. 2021Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)cited by 38
  2. 2023Learning Neural PDE Solvers with Parameter-Guided Channel Attentioncited by 37
  3. 2021Meta-Auto-Decoder for Solving Parametric Partial Differential Equationscited by 36
  4. 2022Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networkscited by 33
  5. 2023Discovering Causal Relations and Equations from Datacited by 33
  6. 2021Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Datacited by 31
  7. 2021Neural Delay Differential Equationscited by 26
  8. 2023Predicting Ordinary Differential Equations with Transformerscited by 26
  9. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  10. 2021Hybrid FEM-NN models: Combining artificial neural networks with the finite element methodcited by 24
  11. 2022Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learningcited by 24
  12. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  13. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  14. 2022Learning Physics-Informed Neural Networks without Stacked Back-propagationcited by 23
  15. 2022Deep Learning and Symbolic Regression for Discovering Parametric Equationscited by 22
  16. 2021Robust discovery of partial differential equations in complex situationscited by 21
  17. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  18. 2023Machine Learning for Partial Differential Equationscited by 20
  19. 2021DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimizationcited by 19
  20. 2021Physics-informed Spline Learning for Nonlinear Dynamics Discoverycited by 18
  21. 2022Self-Consistency of the Fokker-Planck Equationcited by 18
  22. 2023A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Datacited by 18
  23. 2023Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarkscited by 18
  24. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  25. 2021Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solvercited by 15
  26. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  27. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
  28. 2021Generating Physically-Consistent Satellite Imagery for Climate Visualizationscited by 13
  29. 2022Integration of knowledge and data in machine learningcited by 13
  30. 2022Online Weak-form Sparse Identification of Partial Differential Equationscited by 13
  31. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  32. 2022Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertaintycited by 13
  33. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  34. 2024P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamicscited by 13
  35. 2022Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solverscited by 12
  36. 2023Theory and implementation of inelastic Constitutive Artificial Neural Networkscited by 12
  37. 2022PARC: Physics-Aware Recurrent Convolutional Neural Networks to Assimilate Meso-scale Reactive Mechanics of Energetic Materialscited by 11
  38. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11

50 most cited of 208 All 208 (JSON)