TY - GEN
T1 - Calibrated Adaptive Probabilistic ODE Solvers
AU - Bosch, Nathanael
AU - Hennig, Philipp
AU - Tronarp, Filip
N1 - Publisher Copyright:
Copyright © 2021 by the author(s)
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
M3 - Paper in conference proceeding
AN - SCOPUS:85119141008
T3 - Proceedings of Machine Learning Research
SP - 3466
EP - 3474
BT - International Conference on Artificial Intelligence and Statistics
PB - ML Research Press
T2 - 24th International Conference on Artificial Intelligence and Statistics, AISTATS 2021
Y2 - 13 April 2021 through 15 April 2021
ER -