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Probabilistic ODE Solutions in Millions of Dimensions
Nicholas Krämer · Nathanael Bosch · Jonathan Schmidt · Philipp Hennig

Tue Jul 19 10:35 AM -- 10:40 AM (PDT) @ Room 307
in PM: Gaussian Processes »

Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed implementation schemes behind solving high-dimensional ODEs with a probabilistic numerical algorithm. This has not been possible before due to matrix-matrix operations in each solver step, but is crucial for scientifically relevant problems---most importantly, the solution of discretised partial differential equations. In a nutshell, efficient high-dimensional probabilistic ODE solutions build either on independence assumptions or on Kronecker structure in the prior model. We evaluate the resulting efficiency on a range of problems, including the probabilistic numerical simulation of a differential equation with millions of dimensions.

Keywords: PM: Everything Else · PM: Gaussian Processes

Author Information

Nicholas Krämer (University of Tübingen)
Nathanael Bosch (University of Tübingen)
Jonathan Schmidt (University of Tübingen)
Philipp Hennig (University of Tuebingen)

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