Koopman Theory and Predictive Equivalence: Learning Implicit Models of Complex Systems from Partial Observations

Only a subset of degrees of freedom are typically accessible in the real-world. And so, the proper setting for empirical modeling is that of partially-observed systems. To predict the future behavior with a physics simulation model, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parameterization. Recently, data-driven models have consistently outperformed physics simulations for systems with few observable degrees of freedom (e.g. hydrological systems). Here we provide an operator-theoretic explanation for this empirical success. Delay-coordinate embeddings and their evolution under the Koopman operator implicitly model the effects of the missing degrees of freedom. For complex and high-dimensional systems, data-driven models must accommodate noise, and this naturally leads to the concept of predictive equivalence. We employ a far-from-equilibrium Mori-Zwanzig formalism to show how predictive equivalence and the Koopman operator create a physically-consistent stochastic model for the observed degrees of freedom. This work clarifies the relationship between implicit data-driven models and explicit physics simulations. Our causal state analog forecasting algorithm demonstrates these results in practice on real and synthetic data.