  We propose a multi-fidelity neural network surrogate sampling method for the
uncertainty quantification of physical/biological systems described by ordinary
or partial differential equations. We first generate a set of low/high-fidelity
data by low/high-fidelity computational models, e.g. using coarser/finer
discretizations of the governing differential equations. We then construct a
two-level neural network, where a large set of low-fidelity data are utilized
in order to accelerate the construction of a high-fidelity surrogate model with
a small set of high-fidelity data. We then embed the constructed high-fidelity
surrogate model in the framework of Monte Carlo sampling. The proposed
algorithm combines the approximation power of neural networks with the
advantages of Monte Carlo sampling within a multi-fidelity framework. We
present two numerical examples to demonstrate the accuracy and efficiency of
the proposed method. We show that dramatic savings in computational cost may be
achieved when the output predictions are desired to be accurate within small
tolerances.
