光滑空间中权值可编码神经网络的逼近速度
Approximation Rates for Neural Networks with Encodable Weights in Smoothness Spaces
摘要：
我们研究的必要和充分的复杂性的神经网络近似函数从不同的光滑空间的限制下，可编码的网络权重。基于熵参数，我们首先证明了Besov空间，索伯列夫空间和更多空间中神经网络逼近的非零可编码权重的数量的下界。这些结果对所有充分光滑的激活函数都是有效的。之后，我们提供了一个统一的框架，用于构建具有相当一般的激活函数的神经网络的近似单位分割。这使我们能够通过神经网络来近似局部化的泰勒多项式，并利用Bramble-Hilbert引理。基于我们的框架，我们得到了几乎最优的上界在高阶Sobolev范数。这一工作发展了用神经网络逼近偏微分方程解的理论。
英文摘要：
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower bounds for the number of nonzero encodable weights for neural network approximation in Besov spaces, Sobolev spaces and more. These results are valid for all sufficiently smooth activation functions. Afterwards, we provide a unifying framework for the construction of approximate partitions of unity by neural networks with fairly general activation functions. This allows us to approximate localized Taylor polynomials by neural networks and make use of the Bramble-Hilbert Lemma. Based on our framework, we derive almost optimal upper bounds in higher-order Sobolev norms. This work advances the theory of approximating solutions of partial differential equations by neural networks.
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