  Neural networks are widely deployed models across many scientific disciplines
and commercial endeavors ranging from edge computing and sensing to large-scale
signal processing in data centers. The most efficient and well-entrenched
method to train such networks is backpropagation, or reverse-mode automatic
differentiation. To counter an exponentially increasing energy budget in the
artificial intelligence sector, there has been recent interest in analog
implementations of neural networks, specifically nanophotonic neural networks
for which no analog backpropagation demonstration exists. We design
mass-manufacturable silicon photonic neural networks that alternately cascade
our custom designed "photonic mesh" accelerator with digitally implemented
nonlinearities. These reconfigurable photonic meshes program computationally
intensive arbitrary matrix multiplication by setting physical voltages that
tune the interference of optically encoded input data propagating through
integrated Mach-Zehnder interferometer networks. Here, using our packaged
photonic chip, we demonstrate in situ backpropagation for the first time to
solve classification tasks and evaluate a new protocol to keep the entire
gradient measurement and update of physical device voltages in the analog
domain, improving on past theoretical proposals. Our method is made possible by
introducing three changes to typical photonic meshes: (1) measurements at
optical "grating tap" monitors, (2) bidirectional optical signal propagation
automated by fiber switch, and (3) universal generation and readout of optical
amplitude and phase. After training, our classification achieves accuracies
similar to digital equivalents even in presence of systematic error. Our
findings suggest a new training paradigm for photonics-accelerated artificial
intelligence based entirely on a physical analog of the popular backpropagation
technique.
