J. Eur. Opt. Society-Rapid Publ. 22, 27( 2026) 279
Fig. 3. Classification accuracy and loss as functions of MC iterations for a fixed flip size of 1 1 lm 2 under deterministic and random initial state.
Fig
. 4. Training loss( top panel) and accuracy( bottom panel) versus the number of trials for various flip sizes under deterministic initialization.
loss, reflecting its poor starting point far from an optimal weight configuration. The acceptance rate was initially low, and the network required more iterations before reaching stable convergence. Interestingly, in most trials, the network achieved a higher final accuracy of 96 %, despite showing a transient peak in the very early stage( immediately after launch), followed by a slight decrease over 10 50 iterations before resuming steady improvement. This indicates that although deterministic initialization suffers from a disadvantage at the beginning, the system can overcome its imbalance and eventually reach a strong performance.
Random initialization, where the weights are assigned randomly to ± 1, shortens the computation time and yields to faster early convergence, the accuracy increases immediately, and the loss decreases without delay. In contrast, deterministic initialization requires more steps before any performance progress begins. Despite this efficiency advantage, the random scheme ultimately converges to a lower final accuracy( 95 %) than the deterministic one( 96 %). Comparing both configurations shows that each reaches a stable accuracy, with deterministic initialization converging more slowly but achieving a higher final accuracy, while random initialization accelerates the process with a modest reduction in final accuracy.
Increasing the flip size 1 1 lm 2 to 10 10 lm 2 destabilizes the optimization process, reducing accuracy and causing the loss divergence under deterministic initialization( Fig. 4). The observed instability originates from excessively large domain sizes, which reduce the number of neurons. The resulting performance degradation is therefore not an intrinsic limitation of the MCM-based learning scheme, but a consequence of the underlying update process, where larger flips induce stronger perturbations while simultaneously reducing the neuron counts, whereas smaller flips allow finer exploration of the configuration space and promote more stable convergence. Consequently, this behavior reflects an algorithmic trade-off between update step size, neurons number, and convergence stability, rather than a fundamental constraint of the method.
Figure 5 displays theoretical results for a sample input, obtained after optimization from a random initial state using a flip size of 1 1 lm 2, and 80,000 MC iterations. The input image( a) is mapped through the trained domain distribution, revealing the magnetic domain pattern in the hidden layer( b), the output intensity( c), the masked detector-plane intensity( d), along with the corresponding classifier response. For a given input digit‘ 5’, the classifier produces the highest activation for the correct label, matching the input, confirming that MCM effectively encodes input digits into distinct magnetic domain configurations for accurate digits recognition.
3.2
Magnetic domain evolution
Figures 6 and 7 present the magnetic domain patterns obtained after training with different flip sizes for deterministic and random initialization. Small flip sizes( panels a – c) produce well-defined magnetic domain patterns, indicative of stable learning dynamics, whereas larger flip sizes( panels d – e) progressively degrade the pattern resolution, reflecting