  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.
