Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features
This addresses the problem of efficiently modeling Hamiltonian systems for researchers in physics and machine learning, but it is incremental as it builds on existing kernel methods with specific modifications.
The paper tackles learning Hamiltonian dynamics from limited noisy data by proposing a method using reproducing kernel Hilbert spaces with odd symplectic kernels and random feature approximations, demonstrating improved prediction accuracy and data efficiency in simulations for three Hamiltonian systems.
A method for learning Hamiltonian dynamics from a limited and noisy dataset is proposed. The method learns a Hamiltonian vector field on a reproducing kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in particular, odd Hamiltonian vector fields. This is done with a symplectic kernel, and it is shown how the kernel can be modified to an odd symplectic kernel to impose the odd symmetry. A random feature approximation is developed for the proposed odd kernel to reduce the problem size. The performance of the method is validated in simulations for three Hamiltonian systems. It is demonstrated that the use of an odd symplectic kernel improves prediction accuracy and data efficiency, and that the learned vector fields are Hamiltonian and exhibit the imposed odd symmetry characteristics.