This paper demonstrates the application of Bayesian Artificial Neural Networks to Ordinary Differential Equation (ODE) inverse problems. We consider the case of estimating an unknown chaotic dynamical system transition model from state observation data. Inverse problems for chaotic systems are numerically challenging as small perturbations in model parameters can cause very large changes in estimated forward trajectories. Bayesian Artificial Neural Networks can be used to simultaneously fit a model and estimate model parameter uncertainty. Knowledge of model parameter uncertainty can then be incorporated into the probabilistic estimates of the inferred system's forward time evolution. The method is demonstrated numerically by analysing the chaotic Sprott B system. Observations of the system are used to estimate a posterior predictive distribution over the weights of a parametric polynomial kernel Artificial Neural Network. It is shown that the proposed method is able to perform accurate time predictions. Further, the proposed method is able to correctly account for model uncertainties and provide useful prediction uncertainty bounds.