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.