Modeling Discretization Uncertainty in the Solution of Differential Equations
In many areas of applied science, the time and space evolution of variables can be naturally described by differential equation models, which define states implicitly as functions of their own rates of change. Modeling and inference require an explicit representation of the states (the solution), which is typically not available in closed form, but may be approximated by a variety of discretization-based numerical methods. We develop a probability model for the systematic uncertainty introduced when systems of analytically intractable ordinary and partial differential equations are discretized. The result is a probability distribution over the space of possible state trajectories, describing our belief about the unknown solution given information generated from the model. As such, our probabilistic approach provides a useful alternative to deterministic numerical integration techniques, especially in cases when models are chaotic, ill-conditioned, or contain unmodelled functional variability. Based on these results, we develop a fully probabilistic Bayesian approach to inference and prediction for intractable differential equation models from experimental data. We further demonstrate our methodology on a number of challenging forward and inverse problems.