  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 of NNs is mainly exploited, which establishes a
relationship mapping boundaries to boundaries. The exploitation of this
property 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 NNs. Notable representations are invertible
residual networks (i-ResNets) and Neural ordinary differential equations
(Neural ODEs). For these NNs, our set-boundary reachability method only needs
to perform reachability analysis on the boundary of the input set. For NNs
which do not feature this property 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.
