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One of the main problems in current artificial neural network engineering is the lack of de­ sign rules for layered neural network topologies, namely how many hidden layers and how many neurons per hidden layer to choose for the network. This paper offers a theoretical basis for ap­ proaching this problem. Formally proven theories are developed which maximize the connectivity of layered neural networks, which use supervised learning, allowing for a maximum potential stor­ age capacity. Due to space limitations, only the results, which are learning rule independent, are published here. Since neural networks process and store information, their (storage) capacity is a key design issue, and since the interconnection strengths (or weights) contain the information of the neural network, the potential (storage) capacity is proportional to the number of weights in the network. Without a-priori knowledge, all the weights of a neural network are considered to have the same potential, that is, the range of their possible values is identical. A fully connected neural network will therefore have a higher potential capacity than any other interconnection sCheme. However, the maximum number of connections depends on the topology of the neural network. Most neural networks are trained using a supervised learning rule. For these networks, the number of neurons of the input and output layer is determined by the application and are therefore considered constants for the calculations of this publication. The total number of neurons in the network (N) is, without loss of generality, also considered a constant, in order to enable comparisons between different interconnection schemes (networks having interlayer, interlayer plus intralayer, interlayer plus supralayer connections, and plenary neural networks (which contain all three kinds)). For each of the different interconnection schemes, the maximum connectivity topology can be stated as a bounded multivariate non-linear integer maximization problem and is solved using an extension of multidimensional Lagrange multipliers. The maximum connectivity topology and max­ imum number of weights for all layered neural networks with given (fixed) input and output layer sizes are given in the table below, where Lmin represents the minimum number of layers, and Lmax the maximum number of layers. For a description of neural network topologies and related termi­ nology see [1), and for the proofs of all the theorems plus a more extensive bibliography, see [2). Results for the case of variable input and output layer sizes can be found in [3). interconn. marimum connectivity topology scheme Lmin Lmax # neurons per layer (NI) marimum number of weights (Wmax)

Connectivity Maximization of Layered Neural Networks for Supervised Learning