Full-field order-reduced Gaussian Process emulators for nonlinear probabilistic mechanics

Abstract

This paper proposes the novel full-field order-reduced Gaussian Processes (GPs) emulators to address the difficult yet underinvestigated problem of quantifying high-dimensional uncertainty on full-field solution. Firstly, we produce the high-dimensional raw dataset, including uncertain material and load inputs and full-field displacement outputs, by the isogeometric Monte Carlo simulator. Secondly, assisted by proper orthogonal decomposition, we freshly convert the map between raw input and output to the map between their reduced basis coefficients, alleviating the limitations of traditional data-driven methods in handling high-dimensional issues. Thirdly, a machine learning Emulator based on Gaussian Process, from the perspective of Bayesian statistics, is built and trained using the reduced basis coefficients. Consequently, we use the proposed emulators to quickly and directly predict the full-field solution to the new input. The most salient characteristics illustrated by the engineering examples are: (1) the emulator is a universal data-driven/machine learning method that is capable of directly mapping from high-dimensional input to the high-dimensional (i.e., 2940) full-field solution. (2) It still produces accurate non-linear functional approximations with a small number of training samples, and offers the confidence interval indicating the level of trust we have in this prediction. Meanwhile, it significantly improves the scalability of traditional Gaussian Process Regression. (3) This emulator is considerably efficient, in particular, the more degrees of freedom and/or complexity (such as nonlinearity involved) the system has, the more efficient it will be.