GOODE: A Gaussian Off-The-Shelf Ordinary Differential Equation Solver

David John, Vincent Heuveline, Michael Schober
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:3152-3162, 2019.

Abstract

There are two types of ordinary differential equations (ODEs): initial value problems (IVPs) and boundary value problems (BVPs). While many probabilistic numerical methods for the solution of IVPs have been presented to-date, there exists no efficient probabilistic general-purpose solver for nonlinear BVPs. Our method based on iterated Gaussian process (GP) regression returns a GP posterior over the solution of nonlinear ODEs, which provides a meaningful error estimation via its predictive posterior standard deviation. Our solver is fast (typically of quadratic convergence rate) and the theory of convergence can be transferred from prior non-probabilistic work. Our method performs on par with standard codes for an established benchmark of test problems.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-john19a, title = {{GOODE}: A {G}aussian Off-The-Shelf Ordinary Differential Equation Solver}, author = {John, David and Heuveline, Vincent and Schober, Michael}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {3152--3162}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/john19a/john19a.pdf}, url = {https://proceedings.mlr.press/v97/john19a.html}, abstract = {There are two types of ordinary differential equations (ODEs): initial value problems (IVPs) and boundary value problems (BVPs). While many probabilistic numerical methods for the solution of IVPs have been presented to-date, there exists no efficient probabilistic general-purpose solver for nonlinear BVPs. Our method based on iterated Gaussian process (GP) regression returns a GP posterior over the solution of nonlinear ODEs, which provides a meaningful error estimation via its predictive posterior standard deviation. Our solver is fast (typically of quadratic convergence rate) and the theory of convergence can be transferred from prior non-probabilistic work. Our method performs on par with standard codes for an established benchmark of test problems.} }
Endnote
%0 Conference Paper %T GOODE: A Gaussian Off-The-Shelf Ordinary Differential Equation Solver %A David John %A Vincent Heuveline %A Michael Schober %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-john19a %I PMLR %P 3152--3162 %U https://proceedings.mlr.press/v97/john19a.html %V 97 %X There are two types of ordinary differential equations (ODEs): initial value problems (IVPs) and boundary value problems (BVPs). While many probabilistic numerical methods for the solution of IVPs have been presented to-date, there exists no efficient probabilistic general-purpose solver for nonlinear BVPs. Our method based on iterated Gaussian process (GP) regression returns a GP posterior over the solution of nonlinear ODEs, which provides a meaningful error estimation via its predictive posterior standard deviation. Our solver is fast (typically of quadratic convergence rate) and the theory of convergence can be transferred from prior non-probabilistic work. Our method performs on par with standard codes for an established benchmark of test problems.
APA
John, D., Heuveline, V. & Schober, M.. (2019). GOODE: A Gaussian Off-The-Shelf Ordinary Differential Equation Solver. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:3152-3162 Available from https://proceedings.mlr.press/v97/john19a.html.

Related Material