  Parameter estimation and associated uncertainty quantification is an
important problem in dynamical systems characterized by ordinary differential
equation (ODE) models that are often nonlinear. Typically, such models have
analytically intractable trajectories which result in likelihoods and posterior
distributions that are similarly intractable. Bayesian inference for ODE
systems via simulation methods require numerical approximations to produce
inference with high accuracy at a cost of heavy computational power and slow
convergence. At the same time, Artificial Neural Networks (ANN) offer
tractability that can be utilized to construct an approximate but tractable
likelihood and posterior distribution. In this paper we propose a hybrid
approach, where Laplace-based Bayesian inference is combined with an ANN
architecture for obtaining approximations to the ODE trajectories as a function
of the unknown initial values and system parameters. Suitable choices of a
collocation grid and customized loss functions are proposed to fine tune the
ODE trajectories and Laplace approximation. The effectiveness of our proposed
methods is demonstrated using an epidemiological system with non-analytical
solutions, the Susceptible-Infectious-Removed (SIR) model for infectious
diseases, based on simulated and real-life influenza datasets. The novelty and
attractiveness of our proposed approach include (i) a new development of
Bayesian inference using ANN architectures for ODE based dynamical systems, and
(ii) a computationally fast posterior inference by avoiding convergence issues
of benchmark Markov Chain Monte Carlo methods. These two features establish the
developed approach as an accurate alternative to traditional Bayesian
computational methods, with improved computational cost.
