# PDE/PDF-informed adaptive sampling for efficient non-intrusive surrogate modelling

## TL;DR

This paper introduces a PDE/PDF-informed adaptive sampling method for non-intrusive surrogate modeling that enhances efficiency and accuracy across various discretization techniques, including neural network PDE solvers.

## Contribution

It proposes a novel refinement measure based on PDE residuals and probability densities, applicable to complex, non-hypercube parameter spaces without relying on traditional discretization methods.

## Key findings

- Effective in constructing accurate surrogates with fewer samples.
- Compatible with neural network PDE solvers and various discretization techniques.
- Demonstrated improved efficiency and generality through numerical examples.

## Abstract

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.