  Approximating functions of a large number of variables poses particular
challenges often subsumed under the term ``Curse of Dimensionality'' (CoD).
Unless the approximated function exhibits a very high level of smoothness the
CoD can be avoided only by exploiting some typically hidden {\em structural
sparsity}. In this paper we propose a general framework for new model classes
of functions in high dimensions. They are based on suitable notions of {\em
compositional dimension-sparsity} quantifying, on a continuous level,
approximability by compositions with certain structural properties. In
particular, this describes scenarios where deep neural networks can avoid the
CoD. The relevance of these concepts is demonstrated for {\em solution
manifolds} of parametric transport equations. For such PDEs
parameter-to-solution maps do not enjoy the type of high order regularity that
helps to avoid the CoD by more conventional methods in other model scenarios.
Compositional sparsity is shown to serve as the key mechanism forn proving that
sparsity of problem data is inherited in a quantifiable way by the solution
manifold. In particular, one obtains convergence rates for deep neural network
realizations showing that the CoD is indeed avoided.
