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.

翻译：逼近高维函数面临着通常被称为“维度灾难”（CoD）的特殊挑战。除非被逼近的函数具有非常高阶的光滑性，否则只有通过挖掘某种通常隐式的结构稀疏性才能避免维度灾难。本文提出了一种用于高维函数新模型类的通用框架。该框架基于适当的组合维数-稀疏性概念，在连续层面上量化了具有特定结构性质的组合逼近性。特别地，这描述了深度神经网络能够避免维度灾难的场景。这些概念的相关性通过参数输运方程的解流形得以证明。对于此类偏微分方程，参数到解的映射并不具备在其他模型场景中通过更传统方法帮助避免维度灾难的高阶正则性。研究表明，组合稀疏性作为关键机制，证明问题数据的稀疏性以可量化的方式继承给解流形。具体而言，深度神经网络实现的收敛速度表明维度灾难确实被避免了。