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J. Eur. Opt. Society-Rapid Publ. 22, 20( 2026)
2.3 Construction of the artificial neural network model
For the multi-parameter calculations involved, a deep neural network( DNN) model will be implemented in MATLAB using the Deep Learning Toolbox [ 52 ]. The presence of hidden layers in addition to the input and output layers qualifies the model as a deep learning framework. Neurons within the hidden layers interact to process input data from multiple perspectives, enabling the generation of diverse and complex output responses.
2.4 Model training
The workflow will include separate training, validation, and test stages. Initially, the model will be trained using the train Network function. Subsequently, error metrics will be evaluated, and multi-parameter optimization will be performed if required.
2.5 Optimization procedure
The trained model will be tasked with predicting the metaatom parameters, geometries, dimensions, and spatial arrangements required to achieve the specified target. Simulations will then be repeated using the predicted parameters to validate the results, and the model will be retrained using the newly generated data.
2.6 Design and analysis of results
The simulation outcomes of the designed structures will be analyzed through graphical representations and physical interpretation, and their consistency with the defined objectives will be assessed.
3 Results and discussion
This study focuses on achieving a selected photonic band in a dispersion less manner through artificial intelligence – assisted design. A direct indicator of a negative refractive index is the group velocity, given by: v g ¼ @ x =@ k. To accommodate a larger number of design configurations, a binary geometry, as illustrated in Figure 1, is considered.
Such a structure comprises 2 64 distinct design possibilities. The relative dielectric permittivity of the material is chosen as e r ¼ 9. Following the execution of frequency-domain simulation tools for 10 000 randomly generated unit-cell geometries, a dataset is constructed. This dataset consists of 10 000 input samples, each containing 64 binary indicators representing the presence or absence of material, and output parameters including the effective refractive index( n eff) and the bandwidth defined by( x max � x min). As the predictive model, fitrgp command is employed. The resulting model yields a coefficient of determination of R 2 = 0.75 for both output parameters.
Considering Figure 2, although the group velocity model generally produces accurate predictions, the narrow range of the output values causes the R 2 scores to deviate significantly from unity. While the prediction – ground truth plot captures the overall trend – particularly at low
Figure 1. Each unit cell is defined by a randomly generated 8 8 square grid, where the black regions represent dielectric material with relative permittivity e r = 9, and the white regions correspond to vacuum with e r = 1.
absolute group velocities – the linear dependence of the residuals on the predicted values constitutes a notable limitation of the model. The right-skewed distribution observed in the residual histogram confirms a tendency toward underestimation, whereas the pronounced deviations revealed by the ratio comparison indicate that a simple noise interpretation is insufficient. These observations suggest that the model remains open to further improvement. In Figure 3 as the second output parameter, namely the bandwidth, the calculations are based on the spectral region within ± 10 % deviation from the center of the spectrum exhibiting a negative effective refractive index. Although the model successfully learns the overall trend, all predictions systematically fall below the true values. The clearly visible linear increase in the residual – prediction plot indicates that the model errors are not random but instead depend on the predicted values, implying that the underlying true relationship is inherently nonlinear. Furthermore, the asymmetry in the residual histogram and the step-like pattern observed in the ratio comparison demonstrate that the errors are not normally distributed and that the model does not adequately capture the noise characteristics. Consequently, while the model is capable of capturing the trend, particularly for narrower bandwidths, it exhibits limitations in representing the nonlinear relationship. This model is therefore also open to further refinement, although it provides a reasonable level of adequacy for predicting the outputs considered within the scope of this study.
The model is tasked with identifying a geometry that yields an effective refractive index of n ¼�2:0 andabandwidth ratio of
f ¼ x max � x h min
i ¼ 0:15; ð1Þ ðx maxþx min Þ 2
while exhibiting a dispersion diagram that is maximally isolated from the second and fourth bands within this frequency interval. The geometry proposed by the model, along with the corresponding dispersion diagram, is presented in Figure 4.