  Neural networks (NNs) are increasingly applied in safety-critical systems
such as autonomous vehicles. However, they are fragile and are often
ill-behaved. Consequently, their behaviors should undergo rigorous guarantees
before deployment in practice. In this paper, we propose a set-boundary
reachability method to investigate the safety verification problem of NNs from
a topological perspective. Given an NN with an input set and a safe set, the
safety verification problem is to determine whether all outputs of the NN
resulting from the input set fall within the safe set. In our method, the
homeomorphism property and the open map property of NNs are mainly exploited,
which establish rigorous guarantees between the boundaries of the input set and
the boundaries of the output set. The exploitation of these two properties
facilitates reachability computations via extracting subsets of the input set
rather than the entire input set, thus controlling the wrapping effect in
reachability analysis and facilitating the reduction of computation burdens for
safety verification. The homeomorphism property exists in some widely used NNs
such as invertible residual networks (i-ResNets) and Neural ordinary
differential equations (Neural ODEs), and the open map is a less strict
property and easier to satisfy compared with the homeomorphism property. For
NNs establishing either of these properties, our set-boundary reachability
method only needs to perform reachability analysis on the boundary of the input
set. Moreover, for NNs that do not feature these properties with respect to the
input set, we explore subsets of the input set for establishing the local
homeomorphism property and then abandon these subsets for reachability
computations. Finally, some examples demonstrate the performance of the
proposed method.
