Learning High-Dimensional Nonparametric Differential Equations via
  Multivariate Occupation Kernel Functions
Victor Rielly Kamel Lahouel Ethan Lew Michael Wells Vicky Haney Bruno Jedynak
Learning a nonparametric system of ordinary differential equations (ODEs)
from $n$ trajectory snapshots in a $d$-dimensional state space requires
learning $d$ functions of $d$ variables. Explicit formulations 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 to learning using the implicit formulation provided by vector-valued
Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral
form, which we subsequently minimize, we derive our learning algorithm. The
minimization problem's solution for the vector field relies on multivariate
occupation kernel functions associated with the solution trajectories. We
validate our approach through experiments on highly nonlinear simulated and
real data, where $d$ may exceed 100. We further demonstrate the versatility of
the proposed method by learning a nonparametric first order quasilinear partial
differential equation.