Dynamical Systems Behavior Prediction: Machine Learning and Koopman Operator

Dynamic systems are commonly represented in local state space, emphasizing state evolution over time. However, this approach poses challenges in analysis, prediction, and control. A more structured alternative is the use of Ordinary Differential Equations (ODEs), which offer a compact and expressive mathematical formulation but are often computationally demanding and difficult to control. To mitigate these limitations, spectral decomposition techniques enable the transformation of nonlinear dynamics into a linear framework, simplifying analysis and control. In this study, we compare two prominent methodologies for achieving such decompositions: Machine Learning (ML) and Koopman Operator (KO) theory. Specifically, we investigate purely data-driven ML, Physics-Informed ML (IML), datadriven KO, and ODE-based KO, analyzing their respective strengths and limitations. Using the mass-spring-damper system as a case study, we explore the effectiveness of these approaches when either only observational data or explicit ODE formulations are available. To the best of our knowledge, no prior work has systematically compared ML and KO-based methods under these conditions. Our findings aim to provide insights into the advantages and trade-offs of data-driven and physics-informed techniques for modeling and controlling dynamical systems.