Paper
PriorIDENT: Prior-Informed PDE Identification from Noisy Data
Authors
Cheng Tang, Hao Liu, Dong Wang
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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.05946v1</id>\n <title>PriorIDENT: Prior-Informed PDE Identification from Noisy Data</title>\n <updated>2026-03-06T06:25:13Z</updated>\n <link href='https://arxiv.org/abs/2603.05946v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.05946v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.NA'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math-ph'/>\n <published>2026-03-06T06:25:13Z</published>\n <arxiv:primary_category term='math.NA'/>\n <author>\n <name>Cheng Tang</name>\n </author>\n <author>\n <name>Hao Liu</name>\n </author>\n <author>\n <name>Dong Wang</name>\n </author>\n </entry>"
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