PDE/PDF-informed adaptive sampling for efficient non-intrusive surrogate modelling
A novel reﬁnement measure for non-intrusive surrogate modelling 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 a PDE residual and probability density function of the uncertain parameters, and excludes parts of the PDE solution that are not used 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 non-hypercube, and has no restrictions on the type of the discretisation method. Therefore, we are not restricted to conventional discretisation techniques, e.g., ﬁnite elements and 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.
|PDE residual, interpolation, uncertainty quantiﬁcation, non-intrusiveness|
van Halder, Y, Sanderse, B, & Koren, B. (2019). PDE/PDF-informed adaptive sampling for efficient non-intrusive surrogate modelling.