  We present a framework based on interval analysis and monotone systems theory
to certify and search for forward invariant sets in nonlinear systems with
neural network controllers. The framework (i) constructs localized first-order
inclusion functions for the closed-loop system using Jacobian bounds and
existing neural network verification tools; (ii) builds a dynamical embedding
system where its evaluation along a single trajectory directly corresponds with
a nested family of hyper-rectangles provably converging to an attractive set of
the original system; (iii) utilizes linear transformations to build families of
nested paralleletopes with the same properties. The framework is automated in
Python using our interval analysis toolbox $\texttt{npinterval}$, in
conjunction with the symbolic arithmetic toolbox $\texttt{sympy}$, demonstrated
on an $8$-dimensional leader-follower system.
