Leveraging KANs For Enhanced Deep Koopman Operator Discovery

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.

Deep Learning Based Dynamics Identification and Linearization of Orbital Problems using Koopman Theory

The study of the Two-Body and Circular Restricted Three-Body Problems in the field of aerospace engineering and sciences is deeply important because they help describe the motion of both celestial and artificial satellites. With the growing demand for satellites and satellite formation flying, fast and efficient control of these systems is becoming ever more important. Global linearization of these systems allows engineers to employ methods of control in order to achieve these desired results. We propose a data-driven framework for simultaneous system identification and global linearization of both the Two-Body Problem and Circular Restricted Three-Body Problem via deep learning-based Koopman Theory, i.e., a framework that can identify the underlying dynamics and globally linearize it into a linear time-invariant (LTI) system. The linear Koopman operator is discovered through purely data-driven training of a Deep Neural Network with a custom architecture. This paper displays the ability of the Koopman operator to generalize to various other Two-Body systems without the need for retraining. We also demonstrate the capability of the same architecture to be utilized to accurately learn a Koopman operator that approximates the Circular Restricted Three-Body Problem.

Computationally Efficient Data-Driven Discovery and Linear Representation of Nonlinear Systems For Control

This work focuses on developing a data-driven framework using Koopman operator theory for system identification and linearization of nonlinear systems for control. Our proposed method presents a deep learning framework with recursive learning. The resulting linear system is controlled using a linear quadratic control. An illustrative example using a pendulum system is presented with simulations on noisy data. We show that our proposed method is trained more efficiently and is more accurate than an autoencoder baseline.