aiqbench analysis
This paper advances kernel extended dynamic mode decomposition (kEDMD) by integrating dictionary learning techniques to automatically optimize kernel parameters and prune redundant components. By applying gradient-based optimization, the method effectively approximates Koopman operators for nonlinear dynamical systems like the Duffing oscillator without manual feature engineering. The work represents a significant methodological improvement in data-driven analysis of complex physical systems through machine learning.
Dictionary learning for Kernel EDMD
arXiv:2604.25572v2 Announce Type: replace-cross Abstract: Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD, we show how to perform gradient-based optimization over the learnable kernel parameters and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky equation, showcasing the method's different strengths.
Abstract:Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD, we show how to perform gradient-based optimization over the learnable kernel parameters and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky equation, showcasing the method's different strengths.
Dynamical Systems (math.DS); Machine Learning (cs.LG)
arXiv:2604.25572 [math.DS]
(or arXiv:2604.25572v2 [math.DS] for this version)
https://doi.org/10.48550/arXiv.2604.25572 Focus to learn more arXiv-issued DOI via DataCite
Chaos:An.Interdisc.Jou.of.Nonlin.Sci. 36 (2026) 103123-1-20
https://doi.org/10.1063/5.0341144 Focus to learn more DOI(s) linking to related resources
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