It is shown that arbitrary locally finite discrete neural networks are
observable (have the shadowing property) in the sense that pseudo-orb
its obtained by small perturbations of an orbit are approximated by ac
tual orbits. The model includes discretizations of analog networks, ar
bitrary cellular automata, and a wide generalization of linear maps on
a one dimensional grid. It follows that the true qualitative behavior
of dynamical systems can be observed to infinite precision on compute
r simulations, despite unavoidable discretization and approximation er
rors.