Classical reduced models are low-rank approximations using a fixed basis designed to
achieve dimensionality reduction of large-scale systems. In this work, we introduce reduced
deep networks, a generalization of classical reduced models formulated as deep neural
networks. We prove depth separation results showing that reduced deep networks
approximate solutions of parametrized hyperbolic partial differential equations with
approximation error $\epsilon $ with $\mathcal {O}(|\log (\epsilon)|) $ degrees of freedom …