Abstract Deep learning has been a powerful tool in various fields. Faced with the limits of the current electronics platform, optical neural networks (ONNs) based on Si programmable photonic integrated circuits (PICs) have attracted considerable attention as a novel deep-learning scheme with optical-domain matrix-vector multiplication (MVM). However, most of the proposed Si programmable PICs for ONNs have several drawbacks, such as low scalability, high power consumption, and lack of frameworks for training. To address these issues, we have proposed a microring resonator (MRR) crossbar array as a Si programmable PIC for an ONN. In this Article, we present a prototype of a fully integrated 4 × 4 MRR crossbar array for MVM and demonstrate a simple image classification task using this chip. Moreover, we propose on-chip backpropagation using the transpose matrix operation of the MRR crossbar array, enabling the on-chip training of the ONN. The proposed ONN scheme can establish a scalable, power-efficient deep-learning accelerator for both inference and training tasks.