论文标题
在非线性模型降低中减少深网的深度分离：非线性双曲线问题中的蒸馏冲击波
Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems
论文作者
论文摘要
经典还原模型是使用固定基础旨在实现大规模系统的维度降低的固定基础的低级近似值。在这项工作中，我们介绍了简化的深网，这是对深度神经网络的经典还原模型的概括。我们证明了深度分离结果表明，减少的深网近似近似偏微分方方程的近似解决方案，近似误差$ε$，带有$ \ MATHCAL {O}（| \ log（ε）|）$自由度，即使在非线性设置中，解决方案则在溶液中展示了冲击波。我们还表明，经典的减少模型通过在相关的Kolmogorov $ n $ widths上建立下限来实现指数差的近似率。
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 $ε$ with $\mathcal{O}(|\log(ε)|)$ degrees of freedom, even in the nonlinear setting where solutions exhibit shock waves. We also show that classical reduced models achieve exponentially worse approximation rates by establishing lower bounds on the relevant Kolmogorov $N$-widths.
