Complex-Valued Neural Networks: An Introduction

Complex-valued neural networks deal with complex-valued data with complex-number weights and complex-valued neuron-activation functions. George M. Gerogiou describes clearly in the Foreword the necessity of the complex-valued networks. In this introductory short chapter, we discuss how they are or can be useful and effective. We begin with the role of i ≡ √ −1 in the quantum mechanics. According to the quantum mechanics, the motion of an electron is related to the Schrödinger equation: i¯ h ∂Ψ(r, t) ∂t = − ¯ h 2 2m ∇ 2 Ψ(r, t) + V (r)Ψ(r, t) (1) where Ψ(r, t) is the electron's wave function in terms of position r and time t, and ¯ h, m, V (r, t) and ∇ denote Plank constant divided by 2π, electron mass, potential function and spatial differential operator, respectively. The probabilistic interpretation argues that the squared absolute value of the solution |Ψ| 2 is the probability density of electron existence. The probability is related to ensemble average. However, realistically , through repetitive or long-term experiment of electron observation in an ergodic condition, we find that the electrons obey the probability |Ψ| 2. The equation represents experimental results successfully .