Koopman operator-based discussion on partial observation in stochastic systems

It is sometimes difficult to achieve a complete observation for a full set of observables, and partial observations are necessary. For deterministic systems, the Mori-Zwanzig formalism provides a theoretical framework for handling partial observations. Recently, data-driven algorithms based on the Koopman operator theory have made significant progress, and there is a discussion to connect the Mori-Zwanzig formalism with the Koopman operator theory. In this work, we discuss the effects of partial observation in stochastic systems using the Koopman operator theory. The discussion clarifies the importance of distinguishing the state space and the function space in stochastic systems. Even in stochastic systems, the delay-embedding technique is beneficial for partial observation, and several numerical experiments show a power-law behavior of error with respect to the amplitude of the additive noise. We also discuss the relation between the exponent of the power-law behavior and the effects of partial observation.

💡 Research Summary

This paper addresses the challenging problem of partial observation in stochastic dynamical systems by leveraging the Koopman operator framework. While the Mori‑Zwanzig formalism has long provided a theoretical tool for handling unobserved variables in deterministic settings, its direct extension to stochastic differential equations (SDEs) is non‑trivial. The authors first clarify a conceptual distinction that is often blurred in the literature: the state space (the set of possible realizations of the system) and the function space (the space of observables defined on the state space). By underlining state vectors and using separate notation for observable functions, they prevent the common confusion that arises when one treats the observable as if it were a state component.

The paper reviews the Koopman operator for deterministic systems, emphasizing that the operator acts linearly on observables even when the underlying dynamics are nonlinear. Using the Extended Dynamic Mode Decomposition (EDMD), a finite‑dimensional matrix approximation of the Koopman operator is constructed from snapshot pairs ((x_n, y_n)) and a chosen dictionary of basis functions. This data‑driven approach is then linked to the Mori‑Zwanzig projection, where the generalized Langevin equation emerges with a memory kernel and a noise term that represents the effect of unobserved variables.

The core contribution lies in extending this reasoning to stochastic systems described by an Itô SDE
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