  In this paper, we are concerned with the quantification of uncertainties that
arise from intra-day oscillations in the demand for natural gas transported
through large-scale networks. The short-term transient dynamics of the gas flow
is modelled by a hierarchy of hyperbolic systems of balance laws based on the
isentropic Euler equations. We extend a novel adaptive strategy for solving
elliptic PDEs with random data, recently proposed and analysed by Lang,
Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas
transport problems. Sample-dependent adaptive meshes and a model refinement in
the physical space is combined with adaptive anisotropic sparse Smolyak grids
in the stochastic space. A single-level approach which balances the
discretization errors of the physical and stochastic approximations and a
multilevel approach which additionally minimizes the computational costs are
considered. Two examples taken from a public gas library demonstrate the
reliability of the error control of expectations calculated from random
quantities of interest, and the further use of stochastic interpolants to,
e.g., approximate probability density functions of minimum and maximum pressure
values at the exits of the network.
