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Gaussian functional regression for linear partial differential equations

机译：线性偏微分方程的高斯泛函回归

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In this paper, we present a new statistical approach to the problem of incorporating experimental observations into a mathematical model described by linear partial differential equations (PDEs) to improve the prediction of the state of a physical system. We augment the linear PDE with a functional that accounts for the uncertainty in the mathematical model and is modeled as a Gaussian process. This gives rise to a stochastic PDE which is characterized by the Gaussian functional. We develop a Gaussian functional regression method to determine the posterior mean and covariance of the Gaussian functional, thereby solving the stochastic PDE to obtain the posterior distribution for our prediction of the physical state. Our method has the following features which distinguish itself from other regression methods. First, it incorporates both the mathematical model and the observations into the regression procedure. Second, it can handle the observations given in the form of linear functionals of the field variable. Third, the method is non-parametric in the sense that it provides a systematic way to optimally determine the prior covariance operator of the Gaussian functional based on the observations. Fourth, it provides the posterior distribution quantifying the magnitude of uncertainty in our prediction of the physical state. We present numerical results to illustrate these features of the method and compare its performance to that of the standard Gaussian process regression. (C) 2015 Elsevier B.V. All rights reserved.
机译：在本文中，我们提出了一种新的统计方法，用于将实验观察结果纳入由线性偏微分方程（PDE）描述的数学模型中，以改善对物理系统状态的预测的问题。我们通过考虑数学模型中不确定性的函数扩展线性PDE，并将其建模为高斯过程。这产生了以高斯函数为特征的随机PDE。我们开发了一种高斯函数回归方法来确定高斯函数的后验均值和协方差，从而解决随机PDE以获得后验分布，以便我们预测物理状态。我们的方法具有以下特征，有别于其他回归方法。首先，它将数学模型和观察结果都纳入回归过程。其次，它可以处理以字段变量的线性函数形式给出的观察结果。第三，该方法是非参数的，因为它提供了一种系统的方法，可以根据观测值最佳地确定高斯函数的先验协方差算子。第四，它提供了后验分布，量化了我们对物理状态的预测中不确定性的大小。我们提供数值结果来说明该方法的这些特征，并将其性能与标准高斯过程回归的性能进行比较。 （C）2015 Elsevier B.V.保留所有权利。

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