Koopman Operator Based Dynamical Similarity Analysis for Data-driven Quantification of Distance between Dynamics

Quantifying distance between two dynamical neural systems is a fundamental problem in neuroscience and machine learning fields. Neural dynamics are known to possess nonlinear features, which makes comparison between systems difficult. Recently, a promising method to quantify distance between dynamics called Dynamic Similarity Analysis (DSA) is proposed (Ostrow et al., 2023), which measures distance between matrices approximating linear operators defined in time delay embedded space and thus takes nonlinearity into accounts. Although being a strong method, DSA is not free from problems, including obscure interpretability, failure to satisfy the triangle inequality among matrices of different dimensions, and long computational time. To address these problems, we propose a modifiednovel version of DSA. Our proposed DSA measures distance between approximated Koopman operators, which has better interpretability as a linear operator that drives dynamics in a mapped space. The distance measure adopted in our method satisfies the triangle inequality even between matrices of different dimensions. This distance measure also allows extremely fast computational time. We applied our method and DSA to Lorenz system (Lorenz, 1963) of various parameters, and found that our method revealed clusters with respect to parameters and dynamical properties, while DSA failed to do so. With theoretical underpinnings of Koopman operators and matrix distance, we propose our method as an effective method to quantify distance between dynamics.