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.

翻译：传统科学建模通常从固定的、实例化的有效方程出发，随后进行针对特定方程的分析与计算，这一过程在多尺度系统等复杂应用中变得极具挑战性。我们提出一种替代范式，在数学约束的假设类中学习介观动力学。基于广义昂萨格原理，我们引入了一个统一框架，涵盖耗散性和保守性介观动力学。我们在所有学习阶段之前，为该假设类内的所有时空演化方程建立了统一且先验的理论保障，包括全局适定性、渐近稳定性、唯一分解可辨识性以及离散能量耗散性。随后，利用每个问题实例的数据来指导假设类中成员的识别，从而生成准确、鲁棒且可解释的动力学模型。我们通过连续介质偏微分方程模型的数据进行验证，并针对微观链模型（其精确介尺度模型未知）产生的数据开展实证检验。所提出的方法不仅是一种有效的动力学学习器，还能提供对底层物理过程的重要可解释诊断。