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.
翻译:整个工程和科学系统都存在线性系统,最明显的是差异方程式。在许多情况下,系统的强制功能并不为人所知,兴趣在于使用系统的噪音观测来推断强制力,以及其他未知参数。在差异方程式中,强制功能是独立变量的未知功能(通常是时间和空间),可以仿照Gaussian进程(GP ) 。在本文中,我们展示了如何使用以GPs为模型的线性系统连接来有效推导以GPs为模型的强制功能。我们展示了如何精确地计算流性GP(Bayesian ) 参数(如Bayesenel 长度尺度), 在许多情况下,计算率大大低于使用 MMC 方法所需的数值。我们展示了普通和部分差异方程式(GP) 的系统方法, 并展示了基础扩展方法与少量基质矢量的矢量的矢量非常接近真正的强制力。最后,我们展示了如何对非线性模型参数(如Bayesian opimimizations) 进行点估计。