CNN dynamics represents a broader class than PDES

The relationship between Cellular Nonlinear Networks (CNNs) and Partial Differential Equations (PDEs) is investigated. The equivalence between discrete-space CNN models and continuous-space PDE models is rigorously defined. The key role of space discretization is explained. The problem of the equivalence is split into two subproblems: approximation and topological equivalence, that can be explicitly studied for any CNN model. It is known that each PDE can be approximated by a space difference scheme, i.e. a CNN model, that presents a similar dynamic behavior. It is shown, through several examples, that there exist CNN models that are not equivalent to any PDEs, either because they do not approximate any PDE models, or because they have a qualitatively different dynamic behavior (i.e. they are not topologically equivalent to the PDE that they approximate). This proves that the spatio-temporal CNN dynamics is broader than that described by PDEs.