mlsanityData to 7 Oct 2026

PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data

arXiv 2212.04971 ↗First posted 9 Dec 2022cs.LGcited by 20

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Abstract (arXiv)

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, U, to approximate the system response function and a sparse, trainable vector, xi, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of U and xi using a loss function that (1) makes U approximate the system response function, (2) encapsulates the fact that U satisfies a hidden PDE that xi characterizes, and (3) promotes sparsity in xi using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 14 of its 39 references that are in the map. First posted on arXiv 9 Dec 2022.

Builds on

14 of 39 references.

References · 14

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015Automatic differentiation in machine learning: a surveycited by 846
  3. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  4. 2020Physics-informed learning of governing equations from scarce datacited by 167
  5. 2020Weak SINDy For Partial Differential Equationscited by 154
  6. 2020Rational neural networkscited by 75
  7. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 70
  8. 2020DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithmcited by 45
  9. 2021Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)cited by 38
  10. 2021PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learningcited by 24
  11. 2019Data-driven discovery of free-form governing differential equationscited by 16
  12. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
2 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

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Citing papers per year: 2023: 3; 2024: 4; 2025: 6; 2026: 7.

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  1. 2023GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencodercited by 29
  2. 2023Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Datacited by 16
  3. 2024Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured gridcited by 6
  4. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  5. 2025Mechanistic PDE Networks for Discovery of Governing Equationscited by 4
  6. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  7. 2024Data-Driven Discovery of PDEs via the Adjoint Methodcited by 2
  8. 2024Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networkscited by 2
  9. 2025Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivativescited by 2
  10. 2025Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Datacited by 2
  11. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  12. 2025CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equationscited by 1
8 more
  1. 2026Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulationcited by 1
  2. 2026Physics-Informed Neural Networks for Biological 2D+t Reaction-Diffusion Systemscited by 1
  3. 2023A Spectral Approach for Learning Spatiotemporal Neural Differential Equationscited by 0
  4. 2024Taylor-Sensus Network: Embracing Noise to Enlighten Uncertainty for Scientific Datacited by 0
  5. 2026Data Enrichment for Symbolic Regression Using Diffusion Modelscited by 0
  6. 2026On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancementcited by 0
  7. 2026Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observationscited by 0
  8. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0