mlsanityData to 13 Oct 2026

Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Data

arXiv 2309.04699 ↗First posted 9 Sep 2023cs.LGcited by 16

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Abstract (arXiv)

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, U, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.

Key facts. Cited by 16 papers in the map (data through 13 Oct 2026). Builds on 21 of its 48 references that are in the map. First posted on arXiv 9 Sep 2023.

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21 of 48 references.

References · 21

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9 more
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  4. 2022PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Datacited by 20
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  7. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
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Cited by year (2026 to 13 Oct)

Citing papers per year: 2023: 1; 2024: 1; 2025: 3; 2026: 11.

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  1. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  2. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  3. 2024Self-test loss functions for learning weak-form operators and gradient flowscited by 4
  4. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  5. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  6. 2025A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Seriescited by 1
  7. 2026Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulationcited by 1
  8. 2026Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identificationcited by 1
  9. 2026Joint discovery of governing partial differential equations from multi-source datasets by competitive optimizationcited by 1
  10. 2026Symbolic Graph Networks for Robust PDE Discovery from Noisy Sparse Datacited by 0
  11. 2026Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Termscited by 0
  12. 2026Data Enrichment for Symbolic Regression Using Diffusion Modelscited by 0
4 more
  1. 2026Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observationscited by 0
  2. 2026Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selectioncited by 0
  3. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0
  4. 2026Learning PDE Dynamics between Submanifolds Using Green's Observation Operatorscited by 0