Abstract:Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. Data from each problem instance is then used to guide the identification of members within our hypothesis class, giving rise to accurate, robust and interpretable dynamical models. We empirically validate this framework on both data from continuum PDE models as a check, and on data arising from microscopic chain models for which exact meso-scale models are unknown. The proposed approach not only acts as an effective dynamics learner, but also offers vital interpretable diagnostics of the underlying physics.
| Comments: | 38 pages, 10 figures |
| Subjects: | Machine Learning (cs.LG); Dynamical Systems (math.DS) |
| MSC classes: | 00A71, 35M11, 37M10, 82C05, 82C26 |
| Cite as: | arXiv:2605.16211 [cs.LG] |
| (or arXiv:2605.16211v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.16211 arXiv-issued DOI via DataCite (pending registration) |
From: Zhuoyuan Li [view email]
[v1]
Fri, 15 May 2026 17:26:28 UTC (2,951 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。