It is often useful to perform integration over learned functions represented by neural networks. However, this integration is usually performed numerically, as analytical integration over learned functions (especially neural networks) is generally viewed as intractable. In this work, we present a method for representing the analytical integral of a learned function $f$. This allows the exact integral of a neural network to be computed, and enables constrained neural networks to be parametrised by applying constraints directly to the integral. Crucially, we also introduce a method to constrain $f$ to be positive, a necessary condition for many applications (e.g. probability distributions, distance metrics, etc). Finally, we introduce several applications where our fixed-integral neural network (FINN) can be utilised.

翻译：在神经网络表示的学习函数上进行积分运算通常具有实用价值。然而，由于对学习函数（尤其是神经网络）进行解析积分通常被认为难以处理，这类积分通常采用数值方法计算。本研究提出了一种表示学习函数 $f$ 解析积分的方法。该方法可计算神经网络的确切积分，并通过直接对积分施加约束来实现对约束神经网络的参数化。关键的是，我们还引入了一种约束 $f$ 为正的函数约束方法——这是概率分布、距离度量等众多应用的必要条件。最后，我们介绍了固定积分神经网络（FINN）的若干应用场景。