Title: A Convergent RKHS Algorithm for Estimating Nonparametric ODEs
Abstract: Learning nonparametric systems of Ordinary Differential Equations (ODEs) $\dot x = f(t,x)$ from noisy and sparse data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for $f$ for which the solution of the ODE exists and is unique. Learning $f$ consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the $L^2$ distance between $x$ and its estimator. Experiments are provided for simulated data and for the prediction of the Amyloid level in the cortex of aging subjects.
Biosketch: Bruno M. Jedynak received his doctorate in Applied Mathematics and Statistics from the Université Paris Sud in January 1995. He spent a year as a postdoc in the Department of Statistics at the University of Chicago. He was then appointed as assistant professor at the Université des Sciences et Technologies de Lille. From 2003 to 2015, he was a faculty member of the Department of Applied Mathematics and Statistics (AMS) at Johns Hopkins University, first as a visiting professor and then as a research professor. In 2015, he moved to the Fariborz Maseeh Department of Mathematics and Statistics at Portland State University in Oregon where he was appointed as a Maseeh Professor in Mathematical Sciences. He is currently (Fall 2021) visiting the AMS department. He can be contacted at [email protected].
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