  The physical concept of synthetic dimensions has recently been introduced
into optics. The fundamental physics and applications are not yet fully
understood, and this report explores an approach to optical neural networks
using synthetic dimension in time domain, by theoretically proposing to utilize
a single resonator network, where the arrival times of optical pulses are
interconnected to construct a temporal synthetic dimension. The set of pulses
in each roundtrip therefore provides the sites in each layer in the optical
neural network, and can be linearly transformed with splitters and delay lines,
including the phase modulators, when pulses circulate inside the network. Such
linear transformation can be arbitrarily controlled by applied modulation
phases, which serve as the building block of the neural network together with a
nonlinear component for pulses. We validate the functionality of the proposed
optical neural network for the deep learning purpose with examples handwritten
digit recognition and optical pulse train distribution classification problems.
This proof of principle computational work explores the new concept of
developing a photonics-based machine learning in a single ring network using
synthetic dimensions, which allows flexibility and easiness of reconfiguration
with complex functionality in achieving desired optical tasks.
