Scalable Gaussian Process Analysis for Implicit Physics-Based Covariance Models

The performance of Gaussian process analysis can be significantly affected by the choice of the covariance function. Physics-based covariance models provide a systematic way to construct covariance models that are consistent with the underlying physical laws. But the resulting models are still limited by the computational difficulties for large-scale problems. In this study, we propose a new framework combining low-rank approximations and physics-based covariance models to perform both accurate and efficient Gaussian process analysis for implicit models. The proposed approximations interact with the physical model via a black-box forward solver and can achieve quasilinear complexity for Gaussian process regression, maximum likelihood parameter estimations, and approximation of the expected Fisher information matrix when performing uncertainty quantification. We also propose a way to include higher-order terms in the covariance model to account for the nonlinearities. To accomplish the goal, we choose a specific global low-rank approximation of the covariance matrix and use stochastic trace estimators. Our numerical results demonstrate the effectiveness and scalability of the approach, validate the accuracy of maximum likelihood approximations and confidence intervals, and compare the performance favorably with other covariance models.