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J. Eur. Opt. Society-Rapid Publ. 22, 27( 2026)
time-intensive with the present system. Nevertheless, these findings indicate that MCM is intrinsically compatible with real-time physical implementation, whereas BP remains predominantly confined to offline computational optimization. With further optimization of the optical architecture, introducing high-speed polarized camera( HSPC) and digital micromirrors device( DMD), faster magnetic dynamic switching and improved detection, the online training in principle could be significantly accelerated to seconds or even nanoseconds, allowing 60,000 images to be processed within minutes, underscoring the potential of MCM-based optimization in physical and hybrid neuromorphic systems.
4 Summary
Fig
. 8. Accuracy as functions of the hidden-to-output layer separation d2 for different flip sizes, with the input-hidden spacing d1 fixed at 3.0 mm.
We
introduced a new training algorithm based on Monte Carlo Method( MCM) optimization for online learning in magneto-optical diffractive integrated neural network( MO-D 2 NN) applied to image classification. This derivative-free approach enables efficient training in discrete, non-differentiable systems. By avoiding gradient computations, the network achieved consistent 96 % accuracy in classifying handwritten digits. The MCM-based algorithm was successfully applied in an experimental setup, demonstrating the update of magnetic domains in real time. Our study establishes theoretical and practical benefits of MCM optimization, offering a scalable solution to the challenges of gradient free training. While the results are promising, further refinements are needed to accelerate learning, reduce training time, improve efficiency for more complex tasks, and strengthen scalability toward real-time next-generation optical computing.
Funding
This work has been supported by JSPS Grant-in-Aid Scientific Research JP23H04803.
Conflicts of interest The authors declare no conflict of interest.
Figure 9. Loss convergence over 80,000 MCM trials for deterministic( blue) and random( red) initializations. Results are compared with a 5,000-trial BP baseline( inset, top-right).
Magnetic domain was recorded using thermo-magnetic recording system. Detailed fabrication and full experiment procedures are provided in our previous report [ 12, 16 – 18 ]. After each MCM trial, the resulting optical distribution at the output plane was recorded and classification loss was evaluated. Domain updates were retained only upon reduction of the loss, leading to progressive improvement in recognition performance over successive iterations. Processing a single MNIST image required approximately 2 hours over 200 MCM trials under the initial configuration. In an enhanced experimental arrangement presently under active development, 10 images were processed in 4 hours across 1,500 trials, reflecting improved stability and throughput. Despite this progress, scaling to the full dataset remains
Data availability statement
The data supporting the findings of this study can be obtained from the corresponding author upon request.
Author contribution statement
Fatima Zahra Chafi conceptualized, drafted and edited the manuscript, Fatima Zahra Chafi, Tomonao Matsuya, Kanata Watanabe conducted the theoretical study, Hotaka Sakaguchi performed experiments, Takayuki Ishibashi reviewed and guided the analysis of the work, all authors discussed the results and approved the final version.
References
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