Perceptrons with Hebbian Learning Based on Wave Ensembles in Spatially Patterned Potentials

A general scheme to realize a perceptron for hardware neural networks is presented, where multiple interconnections are achieved by a superposition of Schrödinger waves. Spatially patterned potentials process information by coupling different points of reciprocal space. The necessary potential shape is obtained from the Hebbian learning rule, either through exact calculation or construction from a superposition of known optical inputs. This allows implementation in a wide range of compact optical systems, including (1) any nonlinear optical system, (2) optical systems patterned by optical lithography, and (3) exciton-polariton systems with phonon or nuclear spin interactions.

Figure 1

(a) Training phase. A 35 pixel input set corresponding to number one, is activated (blue/left-hand rays) simultaneously with the desired output (red/right-hand rays). The position of the activated output ${\kappa }_1$ identifies the number one. (b) Operation phase. The input set represents a digit with defects and there is no driven field over the output. Scattering with the potential nevertheless gives the maximum output corresponding to $|{\psi }_{{\kappa }_1}{|}^2$ and the number can be recognized.

Figure 2

Pattern recognition of numbers. Top row: Input characters containing several errors. Middle row: Light scattered by the potential is mapped to different outputs. Bottom row: Character recognized by the system.

Figure 3

(a) Nuclear spin induced potential, given by Eq. (19). (b) Phonon-induced weight $w{n}_{ij}$ for a GaAs based system [34]. The plots show the variation with the duration of application of input $k_i$ and output $k_j$, as well as the variation with electron density (a) and polariton density (b).