Verifying safety of neural networks from topological perspectives

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 topological perspectives. 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 first exploited, which establishes rigorous guarantees between the boundaries of the input set and the boundaries of the output set. A homeomorphism is a special case of open maps, and consequently our set-boundary method is considered to be generalized to situations with open map property then². 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 topological property and is easier to satisfy compared with homeomorphisms. For NNs establishing either of these two 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 also 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 our proposed method.