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Verifying the input-output relationships of a neural network so as to achievesome desired performance specification is a difficult, yet important, problemdue to the growing ubiquity of neural nets in many engineering applications. Weuse ideas from probability theory in the frequency domain to provideprobabilistic verification guarantees for ReLU neural networks. Specifically,we interpret a (deep) feedforward neural network as a discrete dynamical systemover a finite horizon that shapes distributions of initial states, and usecharacteristic functions to propagate the distribution of the input datathrough the network. Using the inverse Fourier transform, we obtain thecorresponding cumulative distribution function of the output set, which can beused to check if the network is performing as expected given any random pointfrom the input set. The proposed approach does not require distributions tohave well-defined moments or moment generating functions. We demonstrate ourproposed approach on two examples, and compare its performance to relatedapproaches.

Probabilistic Verification of ReLU Neural Networks via Characteristic Functions