Solution Structure of Multi-layer Neural Networks with Initial Condition

This paper studies the initial value problem of multi-layer cellular neural networks. We demonstrate that the mosaic solutions of such system is topologically conjugated to a new class in symbolic dynamical systems called the path set (Abram and Lagarias in Adv Appl Math 56:109–134, 2014). The topological entropies of the solution, output, and hidden spaces of a multi-layer cellular neural network with initial condition are formulated explicitly. Also, a sufficient condition for whether the mosaic solution space of a multi-layer cellular neural network is independent of initial conditions is addressed. Furthermore, two spaces exhibit identical topological entropy if and only if they are finitely equivalent.