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A. Rupe and V. V. Vesselinov |
ABSTRACT: Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g., hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system's future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel Maximum Entropy and Maximum Caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori-Zwanzig formalism reveal how data-driven models converge to the true dynamics of the observable degrees of freedom. Additionally, Wiener projections show how data-driven models implicitly implement what physics-based models do explicitly, providing a unified modeling framework for predicting complex systems from partial observations.