Implicit Probabilistic Integrators for ODEs
This work advances probabilistic ODE solvers by addressing a known limitation (explicit-only construction), providing a rigorous foundation for implicit variants.
The authors introduce implicit probabilistic integrators for ODEs, extending explicit methods to allow dynamic feedback from future time-steps, and prove their convergence. They demonstrate the method's utility in parameter inference for inverse problems.
We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams-Moulton method. The implicit construction allows for dynamic feedback from the forthcoming time-step, in contrast to previous probabilistic integrators, all of which are based on explicit methods. We begin with a concise survey of the rapidly-expanding field of probabilistic ODE solvers. We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a rigorous proof of its well-definedness and convergence. We discuss the problem of the calibration of such integrators and suggest one approach. We give an illustrative example highlighting the effect of the use of probabilistic integrators - including our new method - in the setting of parameter inference within an inverse problem.