Matching covariance kernels to numerical models in Gaussian process emulation

科学越来越依赖于复杂的数值模型，通常用于求解偏微分方程式，但也依赖于机器学习的“黑箱”模型。如果这样的模型要被用来做出现实世界的决策，那么模型预测需要包括对不确定性的估计。计算这种不确定性的一种方法是使用高斯过程仿真器。这样的模拟器是适合于数值模型的设计实验的统计模型。从本质上讲，我们是在为模型建模。所使用的统计模型是一个高斯过程。这类模型需要一个协方差函数(或核)来描述点之间的协方差如何随距离变化。因为我们不使用任何关于数值模式形式的信息，所以这些技术通常被称为‘黑箱’方法。在这个博士学位中，你将研究不同的协方差核，特别是研究协方差核是否可以与基本偏微分方程的一些性质相匹配。这些方法有时被称为“灰箱”方法，因为我们使用有关底层系统的部分信息。例如，我们知道热方程解是一个带有材料协方差核的高斯过程。然而，我们不知道什么协方差核对应于其他偏微分方程，例如流体流动的Navier-Stokes方程(或它们的线性化版本)。在PHD项目中，您将研究如何将协方差核与基本偏微分方程式进行匹配。您还将探索是否可以使用内核来包含底层模型中存在的约束。例如，我们可能知道输出总是正的，因为输出1总是大于输出2。在我们将看到的大多数例子中，从气候、医疗保健或工程角度来看，方程比一组简单的偏微分方程组要复杂得多，但使用适当的协方差核应该可以提高仿真器的效率。

Science relies increasingly on complex numerical models normally solving partial differentialequations but also 'black box' models from machine learning. If such models are to be used tomake real world decisions the model predictions need to include estimates of uncertainty. Oneway to calculate such uncertainties is the use of Gaussian process emulators. Such emulators arestatistical models fitted to designed experiments of the numerical model. In essence we modelthe model. The statistical model used is a Gaussian process. Such models require a covariancefunction (or kernel) to describe how the covariance between points varies with distance. Becausewe do not use any information about the form of the numerical model these techniques are oftenknown as 'black box' methods. In this PhD scholarship you will be investigating differentcovariance kernels and in particular investigating whether the covariance kernel can be matchedto some properties of underlying partial differential equation. These methods are sometimesreferred to as 'grey box' methods as we use partial information about the underlying system. Asan example we know that the solution to the heat equation is a Gaussian process with a Materncovariance kernel. However we do not know what covariance kernel corresponds to other partialdifferential equations, for example to the Navier-Stokes equations for fluid flow (or a linearisedversion of them). In the PhD project you will investigate how we can match the covariance kernelto the underlying partial differential equations. You will also explore whether we can use thekernels to include constraints present in the underlying model. For example we may know that anoutput is always positive of that output 1 is always larger than output 2. In most of the exampleswe will look at, from climate, healthcare or engineering, the equations are much more complexthan a simple set of partial differential equations but using the appropriate covariance kernelshould improve the efficiency of the emulator.