mlsanityData to 7 Oct 2026

Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning

arXiv 2201.12354 ↗First posted 28 Jan 2022cs.LG cs.AI physics.comp-phcited by 24

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Abstract (arXiv)

There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success in data-driven PDE discovery, the robustness of the existing methods cannot be guaranteed when dealing with low-quality measurement data. To overcome this challenge, we propose a novel physics-encoded discrete learning framework for discovering spatiotemporal PDEs from scarce and noisy data. The general idea is to (1) firstly introduce a novel deep convolutional-recurrent network, which can encode prior physics knowledge (e.g., known PDE terms, assumed PDE structure, initial/boundary conditions, etc.) while remaining flexible on representation capability, to accurately reconstruct high-fidelity data, and (2) perform sparse regression with the reconstructed data to identify the explicit form of the governing PDEs. We validate our method on three nonlinear PDE systems. The effectiveness and superiority of the proposed method over baseline models are demonstrated.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). Builds on 33 of its 56 references that are in the map. First posted on arXiv 28 Jan 2022.

Builds on

33 of 56 references.

References · 33

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21 more
  1. 2017PDE-Net: Learning PDEs from Datacited by 304
  2. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  3. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  4. 2018Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluidscited by 219
  5. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  6. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
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  9. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
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  15. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 8; 2023: 1; 2024: 5; 2025: 8; 2026: 2.

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  1. 2022Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree searchcited by 61
  2. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  3. 2022Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datacited by 18
  4. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  5. 2022SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domaincited by 16
  6. 2022Integration of knowledge and data in machine learningcited by 13
  7. 2024P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamicscited by 13
  8. 2022Physics-informed Deep Super-resolution for Spatiotemporal Datacited by 9
  9. 2024Modeling Spatio-temporal Dynamical Systems with Neural Discrete Learning and Levels-of-Expertscited by 9
  10. 2022CoNSoLe: Convex Neural Symbolic Learningcited by 7
  11. 2022D-CIPHER: Discovery of Closed-form Partial Differential Equationscited by 7
  12. 2023Discovering stochastic partial differential equations from limited data using variational Bayes inferencecited by 6
12 more
  1. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  2. 2025PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systemscited by 5
  3. 2025SlotPi: Physics-informed Object-centric Reasoning Modelscited by 5
  4. 2024Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systemscited by 3
  5. 2025MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulationcited by 3
  6. 2025Physics-Guided Learning of Meteorological Dynamics for Weather Downscaling and Forecastingcited by 3
  7. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  8. 2025PIMRL: Physics-Informed Multi-Scale Recurrent Learning for Burst-Sampled Spatiotemporal Dynamicscited by 1
  9. 2025Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flowscited by 1
  10. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  11. 2024Vision-based Discovery of Nonlinear Dynamics for 3D Moving Targetcited by 0
  12. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0