Probabilistic ODE Solvers with Runge-Kutta Means
This work provides a probabilistic extension to classic ODE solvers, offering richer outputs for researchers and practitioners in numerical analysis and scientific computing, though it is incremental as it builds directly on existing Runge-Kutta methods.
The paper tackles the problem of ordinary differential equation (ODE) solvers by developing probabilistic methods that return probability distributions over solutions instead of point estimates, with results showing these methods match Runge-Kutta solver outputs exactly and have low computational cost.
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.