MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions
This addresses the challenge of scaling differential equation learning to high dimensions for researchers in computational science, though it appears incremental as it builds on existing kernel methods.
The paper tackles the problem of learning nonparametric differential equations from trajectories in high-dimensional state spaces, proposing MOCK, a linear-scaling method that outperforms comparators on some datasets for trajectory and next-point prediction.
Learning a nonparametric system of ordinary differential equations from trajectories in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations often scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach, the multivariate occupation kernel method (MOCK), using the implicit formulation provided by vector-valued reproducing kernel Hilbert spaces. The solution for the vector field relies on multivariate occupation kernel functions associated with the trajectories and scales linearly with the dimension of the state space. We validate through experiments on a variety of simulated and real datasets ranging from 2 to 1024 dimensions. MOCK outperforms all other comparators on 3 of the 9 datasets on full trajectory prediction and 4 out of the 9 datasets on next-point prediction.