  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.
