  Verification and safety assessment of neural network controlled systems
(NNCSs) is an emerging challenge. To provide guarantees, verification tools
must efficiently capture the interplay between the neural network and the
physical system within the control loop. In this paper, a compositional
approach focused on inclusion preserving long term symbolic dependency modeling
is proposed for the analysis of NNCSs. First of all, the matrix structure of
symbolic zonotopes is exploited to efficiently abstract the input/output
mapping of the loop elements through (inclusion preserving) affine symbolic
expressions, thus maintaining linear dependencies between interacting blocks.
Then, two further extensions are studied. Firstly, symbolic polynotopes are
used to abstract the loop elements behaviour by means of polynomial symbolic
expressions and dependencies. Secondly, an original input partitioning
algorithm takes advantage of symbol preservation to assess the sensitivity of
the computed approximation to some input directions. The approach is evaluated
via different numerical examples and benchmarks. A good trade-off between low
conservatism and computational efficiency is obtained.
