Weighted Composition Operators for Learning Nonlinear Dynamics

Area of Science:

• Dynamical Systems
• Operator Theory
• Data Science

Background:

• Operator theoretic methods, including Koopman operators, are prevalent in dynamical systems analysis.
• These methods leverage invariant subspaces and eigenfunctions for linear system modeling.
• Limitations exist when Koopman operators lack exploitable eigenfunctions.

Purpose of the Study:

• To introduce the weighted composition operator as an alternative for dynamical systems study.
• To address limitations of traditional Koopman operator methods.
• To present a new data-driven algorithm for dynamical system analysis.

Main Methods:

• Utilizing weighted composition operators, which are compact across various dynamics and spaces.
• Interacting weighted composition operators with occupation kernels and vector-valued kernels.
• Developing a novel algorithm for data-driven dynamical system modeling.

Main Results:

• Weighted composition operators provide estimations of underlying system dynamics.
• The proposed algorithm facilitates data-driven study of dynamical systems.
• Numerical experiments demonstrate convergence, validating the approach as a proof of concept.

Conclusions:

• The weighted composition operator offers a viable alternative for dynamical systems analysis.
• This operator-based framework can be approximated even without ideal eigenfunctions.
• The presented algorithm provides a new tool for data-driven modeling of complex systems.