KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling

Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at link.

How can we tell if two complex dynamical systems behave similarly over time? This question is vital in fields such as neuroscience, where researchers compare different brain regions, and machine learning, where the goal is to understand how various AI models process information. Traditional methods often fall short when systems behave in nonlinear ways or operate across multiple timescales, including both short-term reactions and long-term trends.We developed KoopSTD, a new tool that captures the essence of how systems evolve by analyzing their behavior in a mathematical space called the Koopman spectrum. This approach allows us to disentangle patterns occurring at different timescales and filter out irrelevant modes that might otherwise lead to misleading conclusions. It is similar to breaking down a complex musical piece into distinct notes and rhythms to better understand its structure.We applied KoopSTD to a wide range of dynamical systems, including classical physical simulations, brain activity, and large language models, and found that it reliably identifies underlying similarities in dynamics. Remarkably, it revealed that brain regions with correlated anatomical features also exhibit similar functional patterns, and that larger AI models tend to have more stable internal dynamics. KoopSTD offers a powerful new way to explore and compare dynamic systems across science and engineering.