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Classical reduced models are low-rank approximations using a fixed basisdesigned to achieve dimensionality reduction of large-scale systems. In thiswork, we introduce reduced deep networks, a generalization of classical reducedmodels formulated as deep neural networks. We prove depth separation resultsshowing that reduced deep networks approximate solutions of parametrizedhyperbolic partial differential equations with approximation error $\epsilon$with $\mathcal{O}(|\log(\epsilon)|)$ degrees of freedom, even in the nonlinearsetting where solutions exhibit shock waves. We also show that classicalreduced models achieve exponentially worse approximation rates by establishinglower bounds on the relevant Kolmogorov $N$-widths.

Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems