Using gradient matching to accelerate parameter inference in nonlinear ordinary differential equations

Ordinary Differential Equations are becoming more widely used throughout all branches of science to model systems of interacting variables. Although researchers can often postulate the structure of the ODEs, there remains a desire to better infer the parameters of these systems. After all, it is these parameters that provide improved understanding of the dynamics involved. Traditionally, parameter inference was done by solving the system of ODEs and assessing fit of the estimated signal with that of the observations. However, nonlinear ODEs often do not permit closed form solutions. Using numerical methods to solve the equations results in prohibitive computational cost, particularly when one adopts a Bayesian approach in sampling parameters from a posterior distribution. The difficulties above have led to the introduction of gradient matching to the parameter inference problem. Instead of quantifying how well the solutions of the ODEs match the data, we quantify how well the derivatives predicted by the ODEs match the derivatives obtained from an interpolant to the data. These methods aim to more efficiently infer the parameters of the equations, but inherent in these procedures is an introduction of bias to the learning problem as we no longer sample based on the exact likelihood function. It is desirable that we obtain a method for parameter inference that is both accurate and efficient, necessitating the involvement of the exact likelihood at some point in the algorithm. Combined with the problems faced in ODE parameter inference, this idea will motivate the main result of this thesis, the introduction of a multiphase scheme in parameter inference that allows us to benefit from the efficiency of the gradient matching likelihood function and the accuracy of the exact likelihood function. The performance of this proposed method is assessed on four benchmark ODE systems, comparing with some standard MCMC sampling techniques from the literature