mlsanityData to 7 Oct 2026

Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

arXiv 2607.29665 ↗First posted 31 Jul 2026cs.LG cs.NA math.NAcited by 0

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Abstract (arXiv)

PDE discovery from sparse observations requires reconstructing a continuous field and selecting the correct differential terms. Our analysis of optimization paths in coupled neural PDE discovery reveals three behaviors: the exact support can persist to the end of training, appear only transiently, or fail to emerge. To decouple equation selection from neural optimization, we develop a freeze-then-select method combining a structured field adapter with Stability-Validated Weak Selection (SVWS). Trained from observations without a PDE residual, the adapter factorizes the field into learned spatial features and temporal coefficients represented by cubic splines. After freezing the field, SVWS identifies recurrent terms across independent weak-form systems, refits candidate supports, and selects the final equation on held-out weak-form systems. Beyond fixed libraries, we apply the same principle to expressions generated by genetic programming and recover the power-law form of an unknown nonlinear diffusion function from sparse, noisy observations. Across all six sparse MDBench regimes, our method attains the highest exact support recovery rate, with its clearest gains over classical and neural baselines on challenging Kuramoto-Sivashinsky dynamics.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 21 of its 43 references that are in the map. First posted on arXiv 31 Jul 2026.

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21 of 43 references.

References · 21

  1. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,214
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  3. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  4. 2017PDE-Net: Learning PDEs from Datacited by 304
  5. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  6. 2020Physics-informed learning of governing equations from scarce datacited by 167
  7. 2020Weak SINDy For Partial Differential Equationscited by 154
  8. 2020SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamicscited by 125
  9. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 70
  10. 2020DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithmcited by 45
  11. 2019DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy datacited by 38
  12. 2018A Unified Framework for Sparse Relaxed Regularized Regression: SR3cited by 36
9 more
  1. 2022PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Datacited by 20
  2. 2021Physics-informed Spline Learning for Nonlinear Dynamics Discoverycited by 18
  3. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
  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. 2020Deep-learning based discovery of partial differential equations in integral form from sparse and noisy datacited by 14
  7. 2022Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertaintycited by 13
  8. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  9. 2026Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulationcited by 1

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