The optical neural network (ONN) based on the Mach-Zehnder interferometer (MZI) has widespread applications in recognition tasks due to its high speed, easy integration, scalability, and insensitivity to external environments. However, errors resulting from manufacturing defects in photonic devices accumulate as the ONN scale increases, consequently diminishing recognition accuracy. To address the decreased accuracy caused by MZI phase errors and beam splitter errors in the MZI-based ONN (MZI-ONN), we introduce a progressive training scheme to reconfigure the phase shift of the MZI feedforward ONN.

Due to the cascaded arrangement of MZIs in MZI-ONN (Fig. 1), the progressive training scheme gradually determines the phase of each column within a certain number of iterations. Based on determining the phase, the phase error and beam splitter error carried by the MZI are considered. After starting the iteration again, the phase value of the undetermined phase shifter is utilized to offset the phase error and beam splitter error carried by the fixed MZI. This training process is repeated until the last column of the grid, and the phase values obtained by progressive training can counteract the inaccuracies caused by imperfect photonic devices, thereby improving the recognition accuracy of MZI-ONN. Importantly, this progressive training scheme reduces inaccuracies caused by optical components without altering the topology of MZI-ONN.

We employ the Neuroptica Python simulation platform to construct a cascaded MZI-ONN and validate the efficacy of the proposed training scheme. The error range of the MZI phase shifter is set between 0.05 and 0.10, with a fixed beam splitter error value of 0.10. Results demonstrate that the proposed progressive training scheme based on the Iris dataset enhances the recognition accuracy of a three-layer 4×4 MZI-ONN from 32.50% to 96.65% (Fig. 5). During the application in the MNIST dataset, the accuracy of three-layer ONNs with grid scales of 4×4, 6×6, 8×8, and 16×16 is elevated by 2.00%, 22.33%, 37.00%, and 36.25%, respectively (Fig. 7), significantly improving the error-resistance performance of the ONN. To substantiate the advantages of the proposed method, we compare the proposed progressive training optimization scheme with traditional genetic algorithm (GA) training, the error correction scheme using a redundant rectangular grid (RRM), and a hardware optimization scheme. Notably, compared with the RRM-based error correction scheme and hardware optimization scheme, the proposed scheme exhibits the capability to conserve more MZI units and detectors. Furthermore, while the traditional GA training scheme enhances the recognition accuracy of the Iris dataset with four features and the MNIST dataset with eight features by 23.10% and 32.40%, respectively, the proposed scheme achieves improvements of 64.15% and 37.00% under the same scale (Table 2). In a comprehensive evaluation, this scheme enhances the recognition accuracy of the ONN without augmenting hardware costs and demonstrates superior error-resistance performance.

We introduce a progressive training scheme designed to alleviate recognition errors in MZI-ONN. The scheme improves the recognition accuracy of the ONN without modifying the topology grid structure and parameters, thus causing no additional hardware costs. To validate the effectiveness of this scheme, we conduct simulations by adopting the Neuroptica Python simulation platform as a proof of concept. The error parameters of photon devices are pre-trained, and the MZI-ONN phase is fixed based on the number of iterations. Subsequent phases are then utilized to compensate for errors introduced by the fixed phase. Simulation analyses are performed on ONNs of scales 4×4, 6×6, 8×8, and 16×16, which demonstrates that the proposed progressive scheme can enhance the recognition accuracy of MZI-ONN by up to 64.15% with an average increase of 39.93%, improving the error-resistant performance of MZI-ONN.