PriorIDENT: Prior-Informed PDE Identification from Noisy Data

TL;DR

PriorIDENT introduces a prior-informed weak-form sparse regression method that effectively identifies PDEs from noisy data by incorporating physical priors and refining candidate libraries, resulting in more accurate and interpretable models.

Contribution

The paper presents a novel PDE identification framework that integrates physics priors into a weak-form sparse regression, improving robustness and interpretability in noisy data scenarios.

Findings

Achieves higher true-positive rates in PDE discovery.

Provides stable coefficient recovery under noise.

Outperforms baseline methods in structure preservation.

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.