🤖 AI Summary
This study addresses the inherent trade-offs among expressiveness, complexity, and data efficiency in modeling nonlinear chaotic systems, where existing data-driven Koopman approaches struggle to achieve both global accuracy and interpretability within finite-dimensional spaces. The authors propose a fuzzy Spectral Region Decomposition (fSRD) framework—a novel, fully automated multi-operator Koopman learning architecture that adaptively constructs local invariant embeddings by integrating fuzzy tree models with spectral learning. This approach enables collaborative, finite-dimensional representations without requiring prior system knowledge. Experimental results demonstrate that fSRD achieves superior prediction accuracy, dynamic interpretability, and data robustness across canonical chaotic systems such as Lorenz and Duffing, as well as high-dimensional real-world datasets, performing effectively in both data-rich and data-scarce regimes.
📝 Abstract
Highly nonlinear chaotic dynamical systems remain difficult to model due to fundamental trade-offs between complexity, expressivity, and data efficiency. Modern machine learning methods achieve strong predictive performance but often rely on a-priori system knowledge or curated data with limited interpretability. Koopman operator theory offers a promising direction via linear representation in an infinite-dimensional observable space. However, many data-driven Koopman methods seek globally valid operators for which useful finite-dimensional spectral embeddings remain difficult to identify under these constraints. To overcome associated limitations, we introduce Fuzzy Spectral Region Decomposition (fSRD), a fully automated learning framework for estimating finite Koopman representation via multiple operators. The proposed method realizes a data-adaptive framework for assembling locally invariant embeddings, termed Invariant Decomposition. fSRD achieves highly accurate linear reconstructions of nonlinear systems while learning finite-dimensional representations of their induced evolution operators, bridging interpretable operator-theoretic models with expressive data-driven sequence learning. These embeddings are adaptively constructed via a global fuzzy tree model, drawing inspiration from fuzzy neural architectures to learn the induced dynamics while prioritizing parsimonious solutions. Empirical results across canonical chaotic systems (e.g., Lorenz and Duffing) and high-dimensional real-world data demonstrate strong predictive accuracy, interpretability, and robust expressivity across data-rich and data-limited regimes, highlighting the method's generality.