We develop a fully Bayesian inferential framework to quantify uncertainty in models defined by general systems of analytically intractable differential equations. This approach provides a statistical alternative to deterministic numerical integration for estimation of complex dynamic systems, and probabilistically characterises the solution uncertainty introduced when models are chaotic, ill-conditioned, or contain unmodelled functional uncertainty.
Viewing solution estimation as an inference problem allows us to quantify numerical uncertainty using the tools of Bayesian function estimation, which may then be propagated through to uncertainty in the model parameters and subsequent predictions. We incorporate regularity assumptions by modelling system states in a Hilbert space with Gaussian measure, and through iterative model-based sampling we obtain a posterior measure on the space of possible solutions, rather than a single deterministic numerical solution that approximately satisfies model dynamics.
We prove some useful properties of this probabilistic solution, propose efficient computational implementation, and demonstrate the methodology on a wide range of challenging forward and inverse problems.