Matching covariance kernels to numerical models in Gaussian process emulation

科学越来越依赖于通常解决偏微分方程的复杂数值模型，也依赖于机器学习的“黑匣子”模型。如果这样的模型被用于现实世界的决策，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模拟器。这种仿真器是拟合数值模型设计实验的统计模型。本质上，我们对模型进行建模。使用的统计模型是高斯过程。这样的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。由于我们不使用任何关于数值模型形式的信息，这些技术通常被称为“黑箱”方法。在这个博士奖学金中，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潜在的偏微分方程的一些性质相匹配。这些方法有时被称为“灰盒”方法，因为我们使用有关底层系统的部分信息。作为一个例子，我们知道热方程的解是一个具有matn协方差核的高斯过程。然而，我们不知道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）。在博士项目中，你将研究如何将协方差核与潜在的偏微分方程相匹配。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，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，方程比一组简单的偏微分方程复杂得多，但使用适当的协方差核应该提高模拟器的效率。

Science relies increasingly on complex numerical models normally solving partial differential equations 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. One way to calculate such uncertainties is the use of Gaussian process emulators. Such emulators are statistical models fitted to designed experiments of the numerical model. In essence we model the model. The statistical model used is a Gaussian process. Such models require a covariance function (or kernel) to describe how the covariance between points varies with distance. Because we do not use any information about the form of the numerical model these techniques are often known as 'black box' methods. In this PhD scholarship you will be investigating different covariance kernels and in particular investigating whether the covariance kernel can be matched to some properties of underlying partial differential equation. These methods are sometimes referred to as 'grey box' methods as we use partial information about the underlying system. As an example we know that the solution to the heat equation is a Gaussian process with a Matern covariance kernel. However we do not know what covariance kernel corresponds to other partial differential equations, for example to the Navier-Stokes equations for fluid flow (or a linearised version of them). In the PhD project you will investigate how we can match the covariance kernel to the underlying partial differential equations. You will also explore whether we can use the kernels to include constraints present in the underlying model. For example we may know that an output is always positive of that output 1 is always larger than output 2. In most of the examples we will look at, from climate, healthcare or engineering, the equations are much more complex than a simple set of partial differential equations but using the appropriate covariance kernelshould improve the efficiency of the emulator.