Parameter Estimation for Delay Differential Equations: A New Galerkin Approximation Framework

Abstract

Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. In this thesis, we present a framework for determining the discrete delays, integral kernels, and other unknown model parameters for these equations from training data through a Galerkin approximation approach proposed by citet{CGLW16}. The adopted Galerkin approximation is based on a particular type of polynomials due to T. H. Koornwinder, which are orthogonal under an inner product with a point mass cite{Koo84}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. Thanks to the analytical nature of the Galerkin approximation, the latter estimation problem can be handled efficiently, even with distributed delays, which is a known computationally challenging scenario. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics.