On the Relationship between CNNs and PDEs

The relationship between Cellular Neural/Nonlinear Networks (CNNs) and Partial Differential Equations (PDEs) is investigated. The equivalence between a discrete-space CNN model and a continuous-space PDE model is rigorously defined. The problem of the equivalence is split into two sub-problems: approximation and topological equivalence, that can be explicitly studied for any CNN models. 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 different dynamic behavior (i.e they are not topologically equivalent to the PDE, that approximate). This proves that the spatio-temporal CNN dynamics is broader than that described by PDEs.