Hypothesis-driven construction of mesoscopic dynamics

2026-05-15

AI Summary

This work addresses the challenge of constructing effective mesoscopic dynamical equations for complex multiscale systems by proposing a hypothesis-driven modeling paradigm grounded in the generalized Onsager principle. Within a class of hypotheses that satisfy prior theoretical constraints—such as global well-posedness and asymptotic stability—the framework unifies the description of dissipative and conservative processes, integrating energy dissipation structure analysis with data-driven techniques to identify concrete models. Validated on both continuous PDE benchmarks and microscopic chain model data, the approach not only accurately reconstructs unknown mesoscopic dynamics but also yields physically interpretable diagnostic insights, thereby achieving a balanced trade-off among accuracy, robustness, and interpretability.

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.

Problem

• mesoscopic dynamics
• multiscale systems
• scientific modeling
• interpretable models
• complex systems

Innovation

• hypothesis-driven modeling
• mesoscopic dynamics
• generalized Onsager principle
• theoretical guarantees
• interpretable dynamics