Bayesian Parameter Estimation of Nonlinear Differential Equations Using Automatic Differentiation

In this paper, an optimization-based Bayesian parameter estimation method is presented. Instead of using the conventional time-consuming integral method, we adopt a novel method, which does not need to solve nonlinear differential equations in the iterative optimization process. To provide the exact direction and magnitude in updating parameter values, we use automatic differentiation (AD) which gives the derivatives as the same accuracy as the analytical ones. Furthermore, we introduce variational inference to represent parameters as probability distributions rather than a single value. Consequently, we show that the uncertainty of the parameter estimates can be quantified with a small cost. We demonstrate the developed method out performs conventional methods and it even works for chaotic systems.