Integrated photonic neural networks provide a promising platform for energy-efficient, high-throughput machine learning with extensive scientific and commercial applications. Photonic neural networks efficiently transform optically encoded inputs using Mach-Zehnder interferometer mesh networks interleaved with nonlinearities. We experimentally trained a three-layer, four-port silicon photonic neural network with programmable phase shifters and optical power monitoring to solve classification tasks using "in situ backpropagation," a photonic analog of the most popular method to train conventional neural networks. We measured backpropagated gradients for phase-shifter voltages by interfering forward- and backward-propagating light and simulated in situ backpropagation for 64-port photonic neural networks trained on MNIST image recognition given errors. All experiments performed comparably to digital simulations ([Formula: see text]94% test accuracy), and energy scaling analysis indicated a route to scalable machine learning.