Neural networks are the state-of-the-art for many approximation tasks in
high-dimensional spaces, as supported by an abundance of experimental evidence.
However, we still need a solid theoretical understanding of what they can
approximate and, more importantly, at what cost and accuracy. One network
architecture of practical use, especially for approximation tasks involving
images, is convolutional (residual) networks. However, due to the locality of
the linear operators involved in these networks, their analysis is more
complicated than for generic fully connected neural networks. This paper
focuses on sequence approximation tasks, where a matrix or a higher-order
tensor represents each observation. We show that when approximating sequences
arising from space-time discretisations of PDEs we may use relatively small
networks. We constructively derive these results by exploiting connections
between discrete convolution and finite difference operators. Throughout, we
design our network architecture to, while having guarantees, be similar to
those typically adopted in practice for sequence approximation tasks. Our
theoretical results are supported by numerical experiments which simulate
linear advection, the heat equation, and the Fisher equation. The
implementation used is available at the repository associated to the paper.

为了更好地理解神经网络对序列近似的精度和成本，本文针对序列近似任务，通过利用离散卷积和有限差分算子之间的联系，构造可以在保证性能的情况下，具有与实践中常用的序列近似任务的相似性的小型卷积神经网络，本文的理论结果得到了数值实验的支持。