Abstract
In this lecture we present a short introduction to the theory of neural networks, in which the mathematical aspects are given special consideration. Static and dynamic aspects of models for pattern retrieval are discussed as well as a statistical mechanical treatment of the Hopfield model in the case where the number of patterns and neurons is large. As a motivation for the study of neural networks we mention the contrast between the fact that von Neumann’s type of sequential or parallel computers are quick in algorithmic computations but slow in performing recognition of patterns, whereas brains (not necessarily human brains) are comparatively slow in algorithmic computations but much quicker in performing pattern recognition. Neural networks (as “caricatures” of biological neuronal networks) are attempts to understand operationally such features of quick recognition of patterns (or signals). In fact they show some efficiency in performing “associative, adaptive memory” (as manifested in retrieval of memories and recognition of simple patterns). The methods used for their study stem from statistical mechanics and the theory of stochastic processes. This is particularly evident in the study of the Hopfield model (Hopfield 1982) to which we shall dedicate most attention in these lectures.
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Albeverio, S., Tirozzi, B. (1997). An Introduction to the Mathematical Theory of Neural Networks. In: Garrido, P.L., Marro, J. (eds) Fourth Granada Lectures in Computational Physics. Lecture Notes in Physics, vol 493. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-14148-9_5
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