Hypothesis-driven construction of mesoscopic dynamics

TL;DR

This paper introduces a hypothesis-driven framework for constructing mesoscopic dynamics that leverages a generalized Onsager principle, providing theoretical guarantees and empirical validation for complex multiscale systems.

Contribution

It presents a novel, mathematically constrained approach to learning mesoscopic models with guarantees, unifying dissipative and conservative dynamics.

Findings

Framework achieves global well-posedness and stability.

Models are accurate, robust, and interpretable.

Validated on continuum PDE and microscopic chain models.

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.