fSRD: Fuzzy Spectral Region Decomposition -- Automated Multi Operator Koopman Representations via an Adaptive Spectral Learning Architecture
2026-07-20 • Machine Learning
Machine Learning
AI summaryⓘ
The authors present a new method called Fuzzy Spectral Region Decomposition (fSRD) to better model chaotic systems, which are usually hard to predict because of their complexity. Their approach breaks down the system into smaller parts and uses multiple simpler operators to build a clearer, linear representation of the system’s behavior. This technique combines ideas from Koopman operator theory and fuzzy modeling to create interpretable and accurate predictions, even when there isn't much data available. They tested fSRD on well-known chaotic examples and real-world data, showing it works well across different situations.
chaotic dynamical systemsKoopman operator theorynonlinear dynamicsspectral embeddingsfuzzy modelinglinear representationinvariant decompositionLorenz systemDuffing oscillatordata-driven modeling
Authors
Charles Bokor, Mark Cary, Denise Morrey, Fabrizio Bonatesta
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.