fSRD: Fuzzy Spectral Region Decomposition – Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture

How can we model highly nonlinear and chaotic dynamical systems using Koopman operator theory without relying on a single, globally valid linear operator or extensive prior system knowledge? The authors address the fundamental trade-off between model expressivity and interpretability, specifically targeting systems where a global linear representation is mathematically impossible due to continuous spectra or multiple attractors.

The authors introduce Fuzzy Spectral Region Decomposition (fSRD), an automated machine learning architecture that learns a collection of local Koopman operators. Instead of forcing a single global embedding, fSRD uses a fuzzy tree model to adaptively partition the state space into locally invariant regions. Within each region, the system approximates the dynamics using a finite-dimensional spectral decomposition. By employing fuzzy logic, the model creates a smooth, data-driven transition between these local operators, effectively assembling a global representation from locally valid linear models.

fSRD demonstrates high predictive accuracy across canonical chaotic systems, such as the Lorenz and Duffing oscillators, as well as high-dimensional real-world datasets. The framework successfully bridges the gap between interpretable operator-theoretic models and expressive sequence learning. Because it does not assume the existence of a global Koopman eigenfunction, it remains robust in regimes where traditional methods (like standard Dynamic Mode Decomposition) fail due to the presence of continuous spectra or non-conjugate invariant sets. The resulting models are parsimonious, prioritizing a minimal set of operators that capture the underlying system structure.

This work provides a systematic, automated alternative to traditional system identification. By moving away from the requirement of a global linear operator, fSRD allows researchers to apply Koopman theory to a much broader class of complex, real-world dynamical systems. It offers a path toward interpretable AI that maintains the rigor of dynamical systems theory while benefiting from the flexibility of modern data-driven architectures.