Gaussian Process Regression for Data Fulfilling Linear Differential Equations with Localized Sources Specialized Gaussian process regression is presented for data that are known to fulfill a given linear differential equation with vanishing or localized sources. The method allows estimation of system parameters as well as strength and location of point sources. It is applicable to a wide range of data from measurement and simulation. The underlying principle is the wellknown invariance of the Gaussian probability distribution under linear operators, in particular differentiation. In contrast to approaches with a generic covariance functionkernel, we restrict the Gaussian process to generate only solutions of the homogeneous part of the differential equation. This requires specialized kernels with a direct correspondence of certain kernel hyperparameters to parameters in the underlying equation and leads to more reliable regression results with less training data. Inhomogeneous contributions from linear superposition of point sources are treated via a linear model over fundamental solutions. Maximum likelihood estimates for hyperparameters and source positions are obtained by nonlinear optimization. For differential equations representing laws of physics the present approach generates only physically possible solutions, and estimated hyperparameters represent physical properties. After a general derivation, modeling of sourcefree data and parameter estimation is demonstrated for Laplacexe2x80x99s equation and the heatdiffusion equation. Finally, the Helmholtz equation with point sources is treated, representing scalar wave data such as acoustic pressure in the frequency domain.

Principal Component Analysis for High Dimension Stochastic Gaussian Process Model Fitting Stochastic Gaussian Process models are widely used in stochastic simulation metamodeling to predict the response of noisy simulations. Often, many realworld engineering problems are highdimensional (more than 10 dimensions), and the Gaussian Process modelsu0027 prediction for such highdimensional inputs are computationally expensive due to multiple inversions of the covariance matrix. Therefore, the xe2x80x9ccurse of dimensionalityxe2x80x9d on these models prevent them from being used in a highdimensional setting. To overcome this problem, a methodology for highdimensional stochastic Gaussian Process modelsu0027 fitting is proposed in this paper. The input data is first projected onto a reduced set of dimensions found by Principal Component Analysis before being used to fit the model. Numerical experiments prove that the method helps significantly in reducing the computation time.