# Adjoint-aided inference of Gaussian process driven differential equations

arXiv: https://arxiv.org/abs/2202.04589 · HTML: https://mlsanity.com/paper/2202.04589/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 58 references that are in the map.
- First posted on arXiv 9 Feb 2022.

## Builds on (most cited first, 8 of 8)

- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Gaussian Process Manifold Interpolation for Probabilistic Atrial Activation Maps and Uncertain Conduction Velocity](https://mlsanity.com/paper/2004.10586/) (2020)
- [Efficient Amortised Bayesian Inference for Hierarchical and Nonlinear Dynamical Systems](https://mlsanity.com/paper/1905.12090/) (2019)
- [Fast Kernel Approximations for Latent Force Models and Convolved Multiple-Output Gaussian processes](https://mlsanity.com/paper/1805.07460/) (2018)
- [Stochastic Variational Bayesian Inference for a Nonlinear Forward Model](https://mlsanity.com/paper/2007.01675/) (2020)
- [Development of hp-inverse model by using generalized polynomial chaos](https://mlsanity.com/paper/1801.03009/) (2018)

## Built on by (most cited first, 2 of 2)

- [Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients](https://mlsanity.com/paper/2212.14319/) (2022)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)

## Abstract (arXiv)

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.