  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 may converge to the true dynamics of the observable
degrees of freedom. Additionally, this framework shows how data-driven models
infer the effects of unobserved degrees of freedom implicitly, in much the same
way that physics models infer the effects explicitly. This provides a unified
implicit-explicit modeling framework for predicting partially-observed systems,
with hybrid physics-informed machine learning methods combining implicit and
explicit aspects.
