GAUSSIAN PROCESS REGRESSION AND CONDITIONAL KARHUNEN-LOÈVE EXPANSION FOR FORWARD UNCERTAINTY QUANTIFICATION AND INVERSE MODELING IN THE PRESENCE OF MEASUREMENT NOISE

Abstract

We propose a new machine learning framework for uncertainty quantification (UQ) and parameter estimation in partial differential equation (PDE) models using sparse noisy measurements of the parameter field. In our approach, the Gaussian process regression (GPR) is used to estimate the distribution of the unknown parameter κ, including mean and variance, conditioned on its measurements. Then, the conditional Karhunen-Loève (KL) expansion of κ and generalized polynomial chaos (gPC) expansion of the state variable u are constructed in terms of the parameter's conditional mean and the eigenfunctions and eigenvalues of the parameter's conditional covariance function. In the forward UQ application, the conditional gPC surrogate is used to estimate the mean and variance of u. Our results show that conditioning reduces the u variance and that the variance decreases with decreasing measurement noise. In the inverse solution, we use the conditional KL and gPC expansions to find a realization of conditional κ distribution that satisfies an appropriate maximum a posteriori minimization problem. We find that the error in the estimated κ decreases with the decreasing observation error.

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