CIRCULAR NODES IN NEURAL NETWORKS

In the usual construction of a neural network, the individual nodes st ore and transmit real numbers that lie in an interval on the real line ; the values are often envisioned as amplitudes. In this article we pr esent a design for a circular node, which is capable of storing and tr ansmitting angular information. We develop the forward and backward pr opagation formulas for a network containing circular nodes. We show ho w the use of circular nodes may facilitate the characterization and pa rameterization of periodic phenomena in general. We describe applicati ons to constructing circular self-maps, periodic compression, and one- dimensional manifold decomposition. We show that a circular node may b e used to construct a homeomorphism between a trefoil knot in R(3) and a unit circle. We give an application with a network that encodes the dynamic system on the limit cycle of the Kuramoto-Sivashinsky equatio n. This is achieved by incorporating a circular node in the bottleneck layer of a three-hidden-layer bottleneck network architecture. Exploi ting circular nodes systematically offers a neural network alternative to Fourier series decomposition in approximating periodic or almost p eriodic functions.