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：对神经网络表示的学习函数进行积分通常具有实用价值。然而，由于对学习函数（尤其是神经网络）的解析积分通常被认为难以处理，此类积分通常采用数值方法计算。本研究提出一种方法，用于表示学习函数 $f$ 的解析积分。这使得神经网络精确积分得以计算，并通过将约束直接应用于积分，实现对约束神经网络的参数化。关键的是，我们同时引入一种方法将 $f$ 约束为正值——这是诸多应用（如概率分布、距离度量等）的必要条件。最后，我们展示了固定积分神经网络（FINN）可被利用的若干应用场景。