Neural Networks Based on the Eigenstates of the Quantum Harmonic Oscillator

The chapter introduces feed-forward neural networks where the hidden units employ orthogonal Hermite polynomials for their activation functions. These neural networks have some interesting properties: (a) the basis functions are invariant under the Fourier transform, subject only to a change of scale, (b) the basis functions are the eigenstates of the quantum harmonic oscillator (QHO), and stem from the solution of Schrodinger’s harmonic equation. The proposed neural networks have performance equivalent to wavelet networks and belong to the general category of nonparametric estimators. They can be used for function approximation, system modelling, image processing and fault diagnosis. These neural networks demonstrate the particle-wave nature of information and give the incentive to analyze significant issues related to this dualism, such as the principle of uncertainty and Balian–Low’s theorem.