  Optical neural networks, employing optical fields and photonic tools to
perform artificial neural network computations, are rapidly advancing and are
generating a broad interest and sparking new applications. We propose a nascent
approach for realizing the optical neural network utilizing a single resonator
network, where the arrival times of optical pulses are interconnected to
construct a synthetic temporal 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 using an example of a complicated wine classification
problem. This proof of principle demonstration opens up an opportunity to
develop a photonics-based machine learning in a single ring network utilizing
the concept of synthetic dimensions. Our approach holds flexibility and
easiness of reconfiguration with potentially complex functionality in achieving
desired optical tasks, pointing towards promisingly perform on-chip optical
computations with further miniaturization.
