Pde/pdf-informed adaptive sampling for efficient nonintrusive surrogate modeling

A novel refinement measure for nonintrusive surrogate modeling of partial differential equations (PDEs) with uncertain parameters is proposed. Our approach uses an empirical interpolation procedure, where the proposed refinement measure is based on both the PDE residual and the probability density function. An important strength is that it ex-cludes parts of the PDE solution that are not required to compute the quantity of interest. The PDE residual used in the refinement measure is computed by using all the partial derivatives that enter the PDE separately. The proposed refinement measure is suited for efficient parametric surrogate construction when the underlying PDE is known, even when the parameter space is nonhypercube, and has no restrictions on the type of the discretization method. Therefore, we are not restricted to conventional discretization techniques, e.g., finite elements or finite volumes, and the proposed method is shown to be effective when used in combination with recently introduced neural network PDE solvers. We present several numerical examples with increasing complexity that demonstrate accuracy, efficiency, and generality of the method.