Calibrated Adaptive Probabilistic ODE Solvers
This work improves the accuracy and efficiency of error estimation for probabilistic ODE solvers, which is important for researchers and practitioners who rely on reliable numerical solutions to differential equations.
This paper addresses the issue of explicit error estimation in probabilistic ODE solvers, where the joint covariance provides a worst-case error estimate but not necessarily an accurate explicit error. The authors introduce and assess several probabilistic calibration methods that, when combined with adaptive step-size selection, lead to descriptive and efficiently computable posteriors, outperforming the Dormand-Prince 4/5 Runge-Kutta method.
Probabilistic solvers for ordinary differential equations assign a posterior measure to the solution of an initial value problem. The joint covariance of this distribution provides an estimate of the (global) approximation error. The contraction rate of this error estimate as a function of the solver's step size identifies it as a well-calibrated worst-case error, but its explicit numerical value for a certain step size is not automatically a good estimate of the explicit error. Addressing this issue, we introduce, discuss, and assess several probabilistically motivated ways to calibrate the uncertainty estimate. Numerical experiments demonstrate that these calibration methods interact efficiently with adaptive step-size selection, resulting in descriptive, and efficiently computable posteriors. We demonstrate the efficiency of the methodology by benchmarking against the classic, widely used Dormand-Prince 4/5 Runge-Kutta method.