Latent functional Gaussian process incorporating output spatial correlations

Expensive physical experiments and time-consuming computer simulations with functional responses have spurred the development of functional surrogate models, which are designed to replace these computer simulations with statistical models. However, conventional functional surrogate modeling often relies on two-stage approaches, which often trigger information loss and fail to adequately capture output spatial correlations. This study proposes a novel one-stage method called Latent Functional Gaussian Process (LFGP) to establish a functional surrogate model. The main idea is to use B-spline basis functions to represent the functional responses, with the coefficients treated as random latent variables following a multivariate Gaussian process. A correlation kernel function called a latent functional B-spline kernel is proposed for LFGP to capture the output spatial correlations in the functional responses. To expedite the proposed one-stage modeling, efficient computational techniques are developed based on the matrix inverse lemma and the Sylvester determinant theorem. Numerical examples demonstrate the superior performance of LFGP over conventional methods. By incorporating closed shape constraints into LFGP, a closed-curve surrogate model is developed and applied to predict dimensional deformations in composite fuselage shape control.