  Nonlinear differential equations (DEs) are used in a wide range of scientific
problems to model complex dynamic systems. The differential equations often
contain unknown parameters that are of scientific interest, which have to be
estimated from noisy measurements of the dynamic system. Generally, there is no
closed-form solution for nonlinear DEs, and the likelihood surface for the
parameter of interest is multi-modal and very sensitive to different parameter
values. We propose a Bayesian framework for nonlinear DE systems. A flexible
nonparametric function is used to represent the dynamic process such that
expensive numerical solvers can be avoided. A sequential Monte Carlo algorithm
in the annealing framework is proposed to conduct Bayesian inference for
parameters in DEs. In our numerical experiments, we use examples of ordinary
differential equations and delay differential equations to demonstrate the
effectiveness of the proposed algorithm. We developed an R package that is
available at \url{https://github.com/shijiaw/smcDE}.
