  Inverse problems exist in a wide variety of physical domains from aerospace
engineering to medical imaging. The goal is to infer the underlying state from
a set of observations. When the forward model that produced the observations is
nonlinear and stochastic, solving the inverse problem is very challenging.
Neural networks are an appealing solution for solving inverse problems as they
can be trained from noisy data and once trained are computationally efficient
to run. However, inverse model neural networks do not have guarantees of
correctness built-in, which makes them unreliable for use in safety and
accuracy-critical contexts. In this work we introduce a method for verifying
the correctness of inverse model neural networks. Our approach is to
overapproximate a nonlinear, stochastic forward model with piecewise linear
constraints and encode both the overapproximate forward model and the neural
network inverse model as a mixed-integer program. We demonstrate this
verification procedure on a real-world airplane fuel gauge case study. The
ability to verify and consequently trust inverse model neural networks allows
their use in a wide variety of contexts, from aerospace to medicine.
