Learning nonparametric ordinary differential equations from noisy data

Description of the problem and related works Fitting a system of nonparametric ordinary differential equations (ODEs) ẋ = f (t, x) to longitudinal data could lead to scientific breakthroughs in disciplines where ODEs or dynamical systems have been used for a long time, including physics, chemistry, and biology, see [1]. By nonparametric, we mean that there is no need to specify the functional form of the vector-field f using a pre-defined finite dimensional parameter. Instead, this force field belongs to a functional space and the number of parameters that characterize this vector field depends on the amount of data available. This provides a great advantage in situations where the form of the vector field is unknown but data is available for learning. The functional spaces considered are Reproducing Kernel Hilbert Spaces (RKHS) [2], allowing for efficient optimization among other desirable properties. A particular difficulty arises when the data is sparse and noisy. This is often the case for longitudinal healthcare data obtained during hospital visits.