A hybrid data driven-physics constrained Gaussian process regression framework with deep kernel for uncertainty quantification

In many practical fields like engineering, finance and physics, we need to explore the effects of uncertainties borne by input or parameters, which is the task of the uncertainty quantification (UQ). In a specific UQ problem, if the system is represented as stochastic partial differential equations, then stochastic Galerkin methods like generalised polynomial chaos (gPC) [29] can be used to obtain statistics of the distribution of the unknown solution. As long as a numerical or analytical solver for the system is available, the renowned Monte Carlo (MC) method [8] can be applied to obtain a distribution of the quantities of interest. However, that is usually prohibitively expensive due to the slow convergence of the Monte Carlo method, despite the fact that the running of the computer code for solving some complicated systems one single time might take days. Thus, to some extent we may be willing to compromise accuracy to reduce running time through formulating a cheap surrogate [2].