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PriorIDENT: Prior-Informed PDE Identification from Noisy Data

Cheng Tang, Hao Liu, Dong Wang

TL;DR

A prior-informed weak-form sparse-regression framework that resolves both issues by refining the dictionary before regression and shifting derivatives onto smooth test functions, demonstrating that compact structural priors, when combined with weak formulations, provide a robust and unified route to physically faithful PDE identification from noisy data.

Abstract

Identifying governing partial differential equations (PDEs) from noisy spatiotemporal data remains challenging due to differentiation-induced noise amplification and ambiguity from overcomplete libraries. We propose a prior-informed weak-form sparse-regression framework that resolves both issues by refining the dictionary before regression and shifting derivatives onto smooth test functions. Our design encodes three compact physics priors-Hamiltonian (skew-gradient and energy-conserving), conservation-law (flux-form with shared cross-directional coefficients), and energy-minimization (variational, dissipative)-so that all candidate features are physically admissible by construction. These prior-consistent libraries are coupled with a subspace-pursuit pipeline enhanced by trimming and residual-reduction model selection to yield parsimonious, interpretable models. Across canonical systems-including Hamiltonian oscillators and the three-body problem, viscous Burgers and two-dimensional shallow-water equations, and diffusion and Allen--Cahn dynamics-our method achieves higher true-positive rates, stable coefficient recovery, and structure-preserving dynamics under substantial noise, consistently outperforming no-prior baselines in both strong- and weak-form settings. The results demonstrate that compact structural priors, when combined with weak formulations, provide a robust and unified route to physically faithful PDE identification from noisy data.

PriorIDENT: Prior-Informed PDE Identification from Noisy Data

TL;DR

A prior-informed weak-form sparse-regression framework that resolves both issues by refining the dictionary before regression and shifting derivatives onto smooth test functions, demonstrating that compact structural priors, when combined with weak formulations, provide a robust and unified route to physically faithful PDE identification from noisy data.

Abstract

Identifying governing partial differential equations (PDEs) from noisy spatiotemporal data remains challenging due to differentiation-induced noise amplification and ambiguity from overcomplete libraries. We propose a prior-informed weak-form sparse-regression framework that resolves both issues by refining the dictionary before regression and shifting derivatives onto smooth test functions. Our design encodes three compact physics priors-Hamiltonian (skew-gradient and energy-conserving), conservation-law (flux-form with shared cross-directional coefficients), and energy-minimization (variational, dissipative)-so that all candidate features are physically admissible by construction. These prior-consistent libraries are coupled with a subspace-pursuit pipeline enhanced by trimming and residual-reduction model selection to yield parsimonious, interpretable models. Across canonical systems-including Hamiltonian oscillators and the three-body problem, viscous Burgers and two-dimensional shallow-water equations, and diffusion and Allen--Cahn dynamics-our method achieves higher true-positive rates, stable coefficient recovery, and structure-preserving dynamics under substantial noise, consistently outperforming no-prior baselines in both strong- and weak-form settings. The results demonstrate that compact structural priors, when combined with weak formulations, provide a robust and unified route to physically faithful PDE identification from noisy data.
Paper Structure (26 sections, 64 equations, 11 figures, 2 tables)

This paper contains 26 sections, 64 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview of the PDE identification framework
  3. Prior-informed dictionary refinement for PDE identification
  4. Proposed framework
  5. Hamiltonian prior: skew-gradient structure
  6. Dictionary design
  7. Conservation-law prior: flux-form representation
  8. Dictionary design
  9. Dictionary design
  10. Energy-dissipation prior: gradient-flow structure
  11. Dictionary design
  12. Case studies
  13. Hamiltonian prior: skew-gradient structure
  14. Conservation-Law prior: flux form representation
  15. Energy-Dissipation prior: gradient flow structure
  16. ...and 11 more sections

Figures (11)

  • Figure 1: TPR results from 20 repeated identification trials for the harmonic oscillator system with different noise levels $\{0\%, 5\%, 10\%, 15\%, 25\%, 50\% \}$ under four configurations as mentioned above.
  • Figure 2: TPR results from twenty repeated identification trials for the three-body Hamiltonian system with different noise levels $\{0\%, 1\%, 5\%, 10\%, 20\%, 50\%\}$ under four configurations as mentioned above.
  • Figure 3: Comparison of trajectories of Three--Body Problem. Top row: True trajectories. Bottom row: Identified trajectories. Columns correspond to noise levels of 0%, 1%, and 5% respectively.
  • Figure 4: TPR results from twenty repeated experiments for Burgers’ equation identification with different noise levels $\{0\%, 1\%, 5\%, 10\%, 25\%, 50\%, 100\%\}$ under four configurations as mentioned above.
  • Figure 5: TPR results from twenty repeated experiments for 2D SWE identification with different noise levels $\{0\%, 1\%, 5\%, 10\%, 20\%, 50\%\}$ under four configurations as mentioned above.
  • ...and 6 more figures