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Stable Implementation of Probabilistic ODE Solvers

Nicholas Krämer, Philipp Hennig; 25(111):1−29, 2024.


Probabilistic solvers for ordinary differential equations (ODEs) provide efficient quantification of numerical uncertainty associated with the simulation of dynamical systems. Their convergence rates have been established by a growing body of theoretical analysis. However, these algorithms suffer from numerical instability when run at high order or with small step sizes---that is, exactly in the regime in which they achieve the highest accuracy. The present work proposes and examines a solution to this problem. It involves three components: accurate initialisation, a coordinate change preconditioner that makes numerical stability concerns step-size-independent, and square-root implementation. Using all three techniques enables numerical computation of probabilistic solutions of ODEs with algorithms of order up to 11, as demonstrated on a set of challenging test problems. The resulting rapid convergence is shown to be competitive with high-order, state-of-the-art, classical methods. As a consequence, a barrier between analysing probabilistic ODE solvers and applying them to interesting machine learning problems is effectively removed.

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