  Increasing the layer number of on-chip photonic neural networks (PNNs) is
essential to improve its model performance. However, the successively cascading
of network hidden layers results in larger integrated photonic chip areas. To
address this issue, we propose the optical neural ordinary differential
equations (ON-ODE) architecture that parameterizes the continuous dynamics of
hidden layers with optical ODE solvers. The ON-ODE comprises the PNNs followed
by the photonic integrator and optical feedback loop, which can be configured
to represent residual neural networks (ResNet) and recurrent neural networks
with effectively reduced chip area occupancy. For the interference-based
optoelectronic nonlinear hidden layer, the numerical experiments demonstrate
that the single hidden layer ON-ODE can achieve approximately the same accuracy
as the two-layer optical ResNet in image classification tasks. Besides, the
ONODE improves the model classification accuracy for the diffraction-based
all-optical linear hidden layer. The time-dependent dynamics property of ON-ODE
is further applied for trajectory prediction with high accuracy.
