Leveraging KANs For Enhanced Deep Koopman Operator Discovery

TL;DR

The paper addresses the data-hungry and computationally intensive challenge of learning Koopman operators for nonlinear dynamics with control by adopting Kolmogorov-Arnold Networks (KANs). It proposes a KANs-based deep Koopman framework that learns the observable map and a finite Koopman operator via EDMD/EDMDc, validated on the pendulum, two-body problem, and pendulum-cart systems. The results show substantial gains in training speed (up to ~31x faster), parameter efficiency (up to ~15x fewer parameters), and predictive accuracy (up to ~1.25x better) while enabling real-time control through LQR using the learned linear model. This work demonstrates the practicality of KANs for efficient, data-efficient, and real-time capable deep Koopman operator learning in aerospace and robotics contexts, with potential for online adaptation and broader nonlinear applications.

Abstract

Multi-layer perceptrons (MLP's) have been extensively utilized in discovering Deep Koopman operators for linearizing nonlinear dynamics. With the emergence of Kolmogorov-Arnold Networks (KANs) as a more efficient and accurate alternative to the MLP Neural Network, we propose a comparison of the performance of each network type in the context of learning Koopman operators with control. In this work, we propose a KANs-based deep Koopman framework with applications to an orbital Two-Body Problem (2BP) and the pendulum for data-driven discovery of linear system dynamics. KANs were found to be superior in nearly all aspects of training; learning 31 times faster, being 15 times more parameter efficiency, and predicting 1.25 times more accurately as compared to the MLP Deep Neural Networks (DNNs) in the case of the 2BP. Thus, KANs shows potential for being an efficient tool in the development of Deep Koopman Theory.

Figures (8)

• Figure 1: Complete schematic of dynamic propagation of the states, including the control input. Note that the KANs block is very simple and non-complex
• Figure 2: KANs learned, self-propagating dynamics prediction given only the same initial condition as the ground truth nonlinear dynamics
• Figure 3: System response to LQR Controller developed using the linear system created with the KANs learned Koopman operator.
• Figure 4: KANs learned, linear propagation of multiple orbits of varying altitudes.
• Figure 5: KANs learned, linear propagation of pendulum-cart system with random excitation as the control input.