Variational Bayes method for ordinary differential equation models

Abstract: Ordinary differential equations (ODEs) are a mathematical model used in many application areas such as climatology, bioinformatics, and chemical engineering with its intuitive appeal to modeling. Despite ODE's wide usage in modeling, the frequent absence of their analytic solutions makes it challenging to estimate ODE parameters from the data, especially when the model has lots of variables and parameters. This paper proposes a Bayesian ODE parameter estimating algorithm which is fast and accurate even for models with many parameters. The proposed method approximates an ODE model with a state-space model based on equations of a numeric solver. It allows fast estimation by avoiding computations of a complete numerical solution in the likelihood. The posterior is obtained by a variational Bayes method, more specifically, the approximate Riemannian conjugate gradient method (Honkela et al. 2010), which avoids samplings based on Markov chain Monte Carlo (MCMC). In simulation studies, we compared the speed and performance of the proposed method with existing methods. The proposed method showed the best performance in the reproduction of the true ODE curve with strong stability as well as the fastest computation, especially in a large model with more than 30 parameters. As a real-world data application, a SIR model with time-varying parameters was fitted to the COVID-19 data. Taking advantage of the proposed algorithm, more than 50 parameters were adequately estimated for each country.