Koopman Universal Neural Dynamic Operator: Achieving Fully Explicit Expression Identification for Nonlinear Dynamical Systems

Dongzhe Zheng, Wenjie Mei

Keywords: neural networks, Koopman operator, universal approximation theorem, system identification, dynamics system modeling

Abstract: Complex nonlinear systems permeate various scientific and engineering domains, presenting significant challenges in accurate modeling and analysis. This paper introduces the Koopman Universal Neural Dynamic Operator (KUNDO), a groundbreaking framework that bridges the gap between data-driven machine learning approaches and traditional mathematical modeling. KUNDO uniquely combines neural networks, Koopman operator theory, and the universal approximation theorem to achieve fully explicit expression identification for complex nonlinear systems. Our framework demonstrates remarkable efficiency in small sample scenarios, overcoming limitations of both classical physical models and black-box machine learning techniques. By learning Koopman-compatible basis functions through neural networks, KUNDO transforms high-dimensional, strongly nonlinear dynamics into interpretable mathematical forms, greatly decreasing the limitations of human selection of basis functions without sacrificing predictive power. We present theoretical analyses of KUNDO's mathematical properties and validate its performance across diverse nonlinear systems. The results showcase KUNDO's potential to revolutionize system identification, offering new avenues for scientific discovery and engineering applications in fields such as climate science, financial modeling, and advanced robotics. This work presents a significant advance towards interpretable AI and data-driven modeling in systems analysis.

Primary Area: learning on time series and dynamical systems

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Submission Number: 6365