Gaussian Process Modeling Using the Principle of Superposition

Partial differential equation (PDE) models of physical systems with initial and boundary conditions are often solved numerically via a computer code called the simulator. To study the dependence of the solution on a functional input, the input is expressed as a linear combination of a finite number of basis functions, and the coefficients of the bases are varied. In such studies, Gaussian process (GP) emulators can be constructed to reduce the amount of simulations required from time-consuming simulators. For linear initial-boundary value problems (IBVPs) with functional inputs as additive terms in the PDE, initial conditions, or boundary conditions, the IBVP solution is theoretically a linear function of the coefficients conditional on all other inputs, which is a result called the principle of superposition. Since numerical errors cause deviation from linearity and nonlinear IBVPs are widely solved in practice, we generalize the result to account for nonlinearity. Based on this generalized result, we propose mean and covariance functions for building GP emulators that capture the approximate conditional linear effect of the coefficients. Numerical simulations demonstrate the substantial improvements in prediction performance achieved with the proposed emulator. Matlab codes for reproducing the results in this article are available in the online supplement.