Calibrated Adaptive Probabilistic ODE Solvers

TL;DR

This paper introduces calibration methods for probabilistic ODE solvers to improve error estimation accuracy, demonstrating their effectiveness and efficiency through numerical experiments and benchmarking against classical methods.

Contribution

It proposes new calibration techniques for probabilistic ODE solvers, enhancing their error estimates and compatibility with adaptive step-size strategies.

Findings

Calibration methods improve error estimate reliability.

Enhanced interaction with adaptive step-size control.

Benchmarking shows competitive performance with classical solvers.

Abstract

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.