Adjoint-aided inference of Gaussian process driven differential equations

Gahungu, Paterne; Lanyon, Christopher W; Álvarez, Mauricio A; Smith, Michael T; Wilkinson, Richard D

Abstract

Linear systems occur throughout engineering and the sciences, most notably as differential equations. In many cases the forcing function for the system is unknown, and interest lies in using noisy observations of the system to infer the forcing, as well as other unknown parameters. In differential equations, the forcing function is an unknown function of the independent variables (typically time and space), and can be modelled as a Gaussian process (GP). In this paper we show how the adjoint of a linear system can be used to efficiently infer forcing functions modelled as GPs, using a truncated basis expansion of the GP kernel. We show how exact conjugate Bayesian inference for the truncated GP can be achieved, in many cases with substantially lower computation than would be required using MCMC methods. We demonstrate the approach on systems of both ordinary and partial differential equations, and show that the basis expansion approach approximates well the true forcing with a modest number of basis vectors. Finally, we show how to infer point estimates for the non-linear model parameters, such as the kernel length-scales, using Bayesian optimisation.

Citation

Gahungu, P., Lanyon, C. W., Álvarez, M. A., Smith, M. T., & Wilkinson, R. D. (2022, November). Adjoint-aided inference of Gaussian process driven differential equations. Presented at NeurIPS 2022: Thirty-sixth Conference on Neural Information Processing Systems, New Orleans, USA and online

• Presentation Conference Type: Edited Proceedings
• Conference Name: NeurIPS 2022: Thirty-sixth Conference on Neural Information Processing Systems
• Start Date: Nov 28, 2022
• End Date: Dec 9, 2022
• Acceptance Date: Dec 5, 2022
• Online Publication Date: Dec 5, 2022
• Publication Date: Dec 6, 2022
• Deposit Date: Oct 13, 2023
• Publicly Available Date: Nov 9, 2023
• Publisher: Massachusetts Institute of Technology Press
• Book Title: Advances in Neural Information Processing Systems 35 (NeurIPS 2022)
• ISBN: 9781713871088
• Public URL: https://nottingham-repository.worktribe.com/output/25956156
• Publisher URL: https://papers.nips.cc/paper_files/paper/2022/hash/6dd16c884345ad63e4708367222410e5-Abstract-Conference.html