About one groupoid associated with the composition of multilayer feedforward neural networks

The authors construct a groupoid whose elements are associated with multilayer feedforward neural networks. This groupoid is called the complete groupoid of the composition of neural networks. Multilayer feedforward neural networks (hereinafter referred to as neural networks) are modelled by defining a special type of tuple. Its components define layers of neurons and structural mappings that specify weights of synaptic connections, activation functions and threshold values. Using the artificial neuron model (that of McCulloch-Pitts) for each such tuple it is possible to define a mapping that models the operation of a neural network as a computational circuit. This approach differs from defining a neural network using abstract automata and related constructions. Modeling neural networks using the proposed method makes it possible to describe the architecture of the network (that is, the network graph, the synaptic weights, etc.). The operation in the full neural network composition groupoid models the composition of two neural networks. A network, obtained as the product of a pair of neural networks, operates on input signals by sequentially applying original networks and contains information about their structure. It is proved that the constructed groupoid is a free.