Data-Driven Response Regime Exploration and Identification for Dynamical Systems

Data-Driven Response Regime Exploration and Identification (DR$^2$EI) is a novel and fully data-driven method for identifying and classifying response regimes of a dynamical system without requiring human intervention. This approach is a valuable tool for exploring and discovering response regimes in complex dynamical systems, especially when the governing equations and the number of response regimes are unknown, and the system is expensive to sample. Additionally, the method is useful for order reduction, as it can be used to identify the most dominant response regimes of a given dynamical system. DR$^2$EI utilizes unsupervised learning algorithms to transform the system's response into an embedding space that facilitates regime classification. An active sequential sampling approach based on Gaussian Process Regression (GPR) is used to efficiently sample the parameter space, quantify uncertainty, and provide optimal trade-offs between exploration and exploitation. The performance of the DR$^2$EI method was evaluated by analyzing three established dynamical systems: the mathematical pendulum, the Lorenz system, and the Duffing oscillator. The method was shown to effectively identify a variety of response regimes with both similar and distinct topological features and frequency content, demonstrating its versatility in capturing a wide range of behaviors. While it may not be possible to guarantee that all possible regimes will be identified, the method provides an automated and efficient means for exploring the parameter space of a dynamical system and identifying its underlying "sufficiently dominant" response regimes without prior knowledge of the system's equations or behavior.