Utilizing the input-output transformation of ultrafast nonlinear pulse propagation in quadratic media, we experimentally construct a multilayer physical neural network to perform both audio and handwritten image classification. We introduce a hybrid in-situ in-silico backpropagation algorithm, physics-aware training, that is resilient to the simulation-reality gap, to train physical neural networks. The methodology for constructing and training physical neural networks applies to generic complex physical systems. To demonstrate its generality, we also built and trained physical neural networks out of analog electronic circuits and multimode mechanical oscillators to perform image classification.
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