Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems
This work addresses the challenge of kernel selection in dynamical system modeling for researchers in machine learning and applied mathematics, representing an incremental improvement over existing Kernel Flows methods.
The paper tackles the problem of learning surrogate models for chaotic dynamical systems by regressing vector fields from observed states, introducing Sparse Kernel Flows to learn optimal kernels from a large dictionary, and applies it to 132 chaotic systems, achieving improved accuracy in kernel selection.
Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term prediction or some other objective for other variants of Kernel Flows). However, this method is limited by the choice of the base kernel. In this paper, we introduce the method of \emph{Sparse Kernel Flows } in order to learn the ``best'' kernel by starting from a large dictionary of kernels. It is based on sparsifying a kernel that is a linear combination of elemental kernels. We apply this approach to a library of 132 chaotic systems.