  Verifying the input-output relationships of a neural network so as to achieve
some desired performance specification is a difficult, yet important, problem
due to the growing ubiquity of neural nets in many engineering applications. We
use ideas from probability theory in the frequency domain to provide
probabilistic verification guarantees for ReLU neural networks. Specifically,
we interpret a (deep) feedforward neural network as a discrete dynamical system
over a finite horizon that shapes distributions of initial states, and use
characteristic functions to propagate the distribution of the input data
through the network. Using the inverse Fourier transform, we obtain the
corresponding cumulative distribution function of the output set, which can be
used to check if the network is performing as expected given any random point
from the input set. The proposed approach does not require distributions to
have well-defined moments or moment generating functions. We demonstrate our
proposed approach on two examples, and compare its performance to related
approaches.
