Learning nonparametric differential equations with operator-valued kernels and gradient matching

Abstract

Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend the scope of gradient matching approaches to nonparametric ODE. A smooth estimate of the solution ODE is built to provide an approximation of the derivative of the ODE solution which is in turn used to learn the nonparametric ODE model. This approach benefits from the flexibility of penalized regression in RKHS allowing for ridge or (structured) sparse regression as well. Very good results are shown on 3 different ODE systems.