We make the case for neural network objects and extend an already existing
neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our
aim will be to show that, yes, indeed, it makes sense to talk about neural
network polynomials, neural network exponentials, sine, and cosines in the
sense that they do indeed approximate their real number counterparts subject to
limitations on certain of their parameters, $q$, and $\varepsilon$. While doing
this, we show that the parameter and depth growth are only polynomial on their
desired accuracy (defined as a 1-norm difference over $\mathbb{R}$), thereby
showing that this approach to approximating, where a neural network in some
sense has the structural properties of the function it is approximating is not
entire intractable.

我们介绍了神经网络对象并扩展了已有的神经网络微积分，目的是证明神经网络多项式、神经网络指数、正弦和余弦确实能够近似其实数对应物，以某些参数 q 和 ε 发生限制。此外，我们还表明参数和深度的增长仅对所需精度（在实数域上定义的 1 - 范数差异）是多项式的，从而证明这种逼近方法中，神经网络在某种程度上具有其所逼近函数的结构特性并非完全无法处理。