Fixed Integral Neural Networks
This work addresses a fundamental limitation in using neural networks for tasks requiring integration, such as in probability modeling or distance metrics, by providing a novel analytical approach.
The authors tackled the problem of performing analytical integration over learned neural network functions, which is typically considered intractable, by introducing a method to represent the exact integral of a neural network, enabling direct application of constraints to the integral and ensuring positivity for applications like probability distributions.
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.