Probabilistic ODE Solutions in Millions of Dimensions

Nicholas Krämer, Nathanael Bosch, Jonathan Schmidt, Philipp Hennig
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:11634-11649, 2022.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-kramer22b, title = {Probabilistic {ODE} Solutions in Millions of Dimensions}, author = {Kr{\"a}mer, Nicholas and Bosch, Nathanael and Schmidt, Jonathan and Hennig, Philipp}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {11634--11649}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/kramer22b/kramer22b.pdf}, url = {https://proceedings.mlr.press/v162/kramer22b.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Probabilistic ODE Solutions in Millions of Dimensions %A Nicholas Krämer %A Nathanael Bosch %A Jonathan Schmidt %A Philipp Hennig %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-kramer22b %I PMLR %P 11634--11649 %U https://proceedings.mlr.press/v162/kramer22b.html %V 162 %X 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.
APA
Krämer, N., Bosch, N., Schmidt, J. & Hennig, P.. (2022). Probabilistic ODE Solutions in Millions of Dimensions. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:11634-11649 Available from https://proceedings.mlr.press/v162/kramer22b.html.

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