mlsanityData to 14 Oct 2026

PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning

arXiv 2111.00998 ↗First posted 1 Nov 2021cs.LGcited by 24

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Abstract (arXiv)

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are sparse and noisy. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.

Key facts. Cited by 24 papers in the map (data through 14 Oct 2026). Builds on 15 of its 36 references that are in the map. First posted on arXiv 1 Nov 2021.

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15 of 36 references.

References · 15

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  3. 2015Automatic differentiation in machine learning: a surveycited by 855
  4. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 235
  5. 2020Physics-informed learning of governing equations from scarce datacited by 167
  6. 2020Weak SINDy For Partial Differential Equationscited by 155
  7. 2017Sparse learning of stochastic dynamic equationscited by 76
  8. 2020Rational neural networkscited by 75
  9. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 71
  10. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 44
  11. 2019DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy datacited by 38
  12. 2019Data-driven discovery of free-form governing differential equationscited by 16
3 more
  1. 2021Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Datacited by 12
  2. 2017Neural network augmented inverse problems for PDEscited by 11
  3. 2020Evolutionary-Based Sparse Regression for the Experimental Identification of Duffing Oscillatorcited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 5; 2024: 4; 2025: 6; 2026: 6.

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  1. 2023GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencodercited by 30
  2. 2024A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modelingcited by 22
  3. 2022PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Datacited by 20
  4. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  5. 2023Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Datacited by 16
  6. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  7. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  8. 2024ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identificationcited by 11
  9. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  10. 2023Discovering stochastic partial differential equations from limited data using variational Bayes inferencecited by 6
  11. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  12. 2024Physics-Constrained Learning for PDE Systems with Uncertainty Quantified Port-Hamiltonian Modelscited by 4
12 more
  1. 2023Data-Driven Autoencoder Numerical Solver with Uncertainty Quantification for Fast Physical Simulationscited by 3
  2. 2024Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systemscited by 3
  3. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  4. 2025Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivativescited by 2
  5. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  6. 2025Plug-and-Play Physics-informed Learning using Uncertainty Quantified Port-Hamiltonian Modelscited by 1
  7. 2025The Machine Learning Approach to Moment Closure Relations for Plasma: A Reviewcited by 1
  8. 2026Joint discovery of governing partial differential equations from multi-source datasets by competitive optimizationcited by 1
  9. 2026WGFINNs: Weak formulation-based GENERIC formalism informed neural networkscited by 0
  10. 2026On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancementcited by 0
  11. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0
  12. 2026See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEscited by 0