Spectrum prediction improvement with phenomenological statistical learning

1 Introduction

Researchers typically analyze photonic systems using one of two problem frameworks, the forward problem and the inverse problem. The forward problem, concerning the acquisition of electromagnetic responses from known structures, can be solved using well-established numerical electromagnetic simulators based on rigorous Maxwell’s equations (Inan and Marshall, 2011; Liu and Fan, 2012). The inverse problem—determining a structure that produces a desired electromagnetic response—remains a challenge (Molesky et al., 2018; Adi et al., 2024; Painter et al., 1999; Knight, 2003; Xu et al., 2008; Eichenfield et al., 2009; Liu et al., 2010; Søndergaard Jensen and Sigmund, 2011).

The inverse problem is commonly addressed through optimization frameworks (Campbell et al., 2019; Newton, 1736; Hestenes and Stiefel, 1952; Donald, 1963; Holland, 1992; Poli et al., 2007; Storn and Price, 1997; Rocca et al., 2011; Hansen et al., 2003). Zagaglia et al. proposed a novel particle swarm optimization (PSO) algorithm to design 2D diffraction gratings composed of semiconductor nanowires (Zagaglia et al., 2022). Wang et al. employed a genetic algorithm to optimize the metasurface geometry of the light-trapping structure (Wang et al., 2013). Vidya et al. use the adjoint gradient method to optimize the surface texture of light-trapping materials (Ganapati et al., 2013). Wang et al. implemented an optimized direct-binary-search (DBS) algorithm to design nanostructures that enhance power conversion efficiency in ultrathin photovoltaic devices (Wang and Menon, 2013). While these algorithms work well across various applications, their main drawback is high computational demand (Guirguis et al., 2019). This is because these optimization algorithms fundamentally depend on repeated electromagnetic simulations—such as finite-difference time-domain (FDTD) or rigorous coupled-wave analysis (RCWA)—which numerically solve Maxwell’s equations (Taflove et al., 2005; Moharam et al., 1995; Cecil Zienkiewicz and Leroy Taylor, 2005).

Statistical learning have revolutionized the simulation and design paradigms in photonic systems research: neural networks are effective in capturing nonlinear relationships in high-dimensional parameter spaces, while Maxwell’s equations provide a rigorous physical foundation for model training and validation in specific problem domains (Ma et al., 2021; Jiang et al., 2021). In a representative study, Shameli et al. demonstrated the effectiveness of neural networks as surrogate models for solving inverse problems in photonic system design (Ali Shameli et al., 2021). Their approach, which replaces computationally intensive numerical simulations with trained neural networks, enables efficient prediction of absorption spectrum for light-trapping structures. This methodology achieves substantial reductions in computational overhead, thereby facilitating the practical implementation of large-scale optimization algorithms. However, this purely data-driven statistical learning approach presents two fundamental limitations. First, while the neural network predicts absorption spectra, it remains unclear whether it has truly captured the underlying physical relationships. Second, there inevitably exists a discrepancy between the neural network’s predictions and results obtained through rigorous computational methods. Consequently, enhancing both the interpretability and predictive accuracy of neural networks emerges as a challenge.

Unlike purely data-driven methods, phenomenological theory uses simple models to describe observed phenomena without requiring detailed physical mechanisms (Steven, 2015). These simplified descriptions make it easier for neural networks to learn meaningful patterns compared to complex first-principles calculations. Building on this advantage, we combine phenomenological theory with machine learning by incorporating physical knowledge into the training process. This integration improves both the interpretability and accuracy of spectrum predictions.

The remainder of this paper is organized as follows. Section 2 introduces the proposed “Resonance-informed model (RIM)” and compares it with the baseline “Black-box model (BBM)”. Section 3 presents the simulation setup and results, followed by a discussion of the findings in Section 4. Finally, Section 5 concludes the paper and outlines potential directions for future research.

2 Methods

In recent decades, light trapping has been extensively studied as a means to enhance light absorption in bulk structures (Yablonovitch, 1982; Joannopoulos et al., 1997; Wang et al., 2012; Garnett and Yang, 2010; Yu et al., 2011; Wang et al., 2014; Li et al., 2020). In our simulation, we optimized the structure of the neural network and utilized it to predict the absorption of light trapping materials, aiming to enhance both the accuracy and interpretability of spectrum predictions.

The framework of the Black-box model is presented in Figure 1a, where structure parameters are the input of the Black-box model. Figure 1b shows an example of the structure parameters in our simulations. The output of the Black-box model is the corresponding absorption spectrum for incident light, with each output value representing the absorption at a specific wavelength in Figure 1c. In our simulation, the wavelengths were sampled from 800 nm to 1,100 nm with a resolution of 3 nm, resulting in a total of 101 wavelengths. The model consists of seven hidden layers which are not shown in Figure 1a, with a ReLU activation function applied after each layer for non-linear processing (Nair and Hinton, 2010). The number of neurons in the seven hidden layers is 64, 128, 256, 512, 1,024, 2,048, and 4,096, respectively. The training employed the L1 loss function. The Adam optimizer was used with an initial learning rate of 0.0003, and the learning rate was decayed exponentially by a factor of 0.95 every 50 epochs. A batch size of 512 was adopted, and the model was trained for 1,000 epochs. All other hyperparameters were set to their default values.

In contrast to the Black-box model, the Resonance-informed model, as shown in Figure 2, consists of three submodels: the Region Model, the Resonance Model, and the Nonresonance Model. All three submodels share identical network architectures and identical structural parameter inputs with the Black-box model in Figure 1a. The Region Model predicts the resonance region probability of the structure at each wavelength. To achieve this, we use statistical methods to determine whether the absorption at a given wavelength lies within the resonance region. During training, for each wavelength, we define Average Absorption aswhere Absorption( ω , i ) is the absorption for the i th light trapping structure at the incident wavelength ω calculated by simulators based on rigorous Maxwell’s equation. N is the total number of light-trapping structure samples in one simulation. The Average Absorption is computed solely from the training set. This value, along with any derived statistical thresholds, is then held fixed and applied to the validation and test sets to prevent data leakage.

A v e r a g e A b s o r p t i o n ω = ∑ i = 1 N A b s o r p t i o n ω , i

If the absorption of a sample at a particular wavelength ω exceeds Average Absorption( ω ), it is considered to be in the resonance region, and the training target for that wavelength is set to 1. Conversely, if the absorption is lower than Average Absorption( ω ), the target is set to 0. This approach is employed to train the Region Model to predict the resonance probability. To ensure that Region Model’s output lies within the range of 0–1, we have added a sigmoid activation function at the final layer of the network (George, 1989).

The Resonance Model and Nonresonance Model, both of which take the structure parameters as input, use the absorption spectrum calculated by simulators as their target outputs but are trained with different weights. During training, we first compute the output of the trained Region Model to obtain the resonance region probability (P). P is then used to assign weights to the Resonance Model directly. The weight for the Nonresonance Model is 1-P. Since the output of the Region Model is close to 1 in the resonance region and close to 0 in the non-resonance region, the Resonance Model is optimized to predict absorption more accurately in the resonance region. The Nonresonance Model, similar to the Resonance Model, is optimized to predict absorption more accurately in the non-resonance region.

For a representative test sample, we illustrate the workflow and intermediate outputs of our Resonance-informed Model in Figure 3. The structure parameters are L = 690 nm, b = 122 nm, h = 167 nm, H = 747 nm. The Region Model outputs a continuous resonance probability P ( λ ) (Figure 3c). A threshold (t = 0.5) is applied to P ( λ ) to generate a binary mask, where regions with P ( λ ) > t are designated as resonant (mask = 1) and others as non-resonant (mask = 0). This mask guides the combination of the Resonance Model and Nonresonance Model predictions to produce the final spectrum (Figures 3d,e). Notably, the threshold t determines the weighting between the two models. When t = 1, the final spectrum degenerates to the Resonance Model’s spectrum, and when t = 0, it degenerates to the Nonresonance Model’s spectrum. Empirically, the results show little variation across a practical range of t (e.g., 0.3–0.7). The absolute error between predicted spectrum and the first-principles reference is computed pointwise and plotted as a red dashed curve (Figures 3b,f).

In each simulation, we randomly generated 6,000 samples for training, 1,600 for validation (model selection), and 400 for testing. The optical absorption spectra of the photonic structures were simulated using the rigorous coupled-wave analysis (RCWA) method via the S 4 simulator (Liu and Fan, 2012), which computes electromagnetic responses by modeling materials with complex refractive indices in a layered geometry. Our neural network model was implemented in the PyTorch framework (Paszke et al., 2019) and trained on an NVIDIA RTX 2080 GPU.

3 Results

We initially tested our model on a two-dimensional triangular textured light-trapping structure. A total of 6,000 training samples were used to train both the Resonance-informed model and the Black-box model for comparison. The variation ranges of L, b, h and H are from 600 to 800 nm, 100 to 200 nm, 100 to 200 nm, and 800 to 1,000 nm, respectively. All parameters were independently and randomly selected within these ranges. As shown in Figure 4a, the light-trapping structure is placed on a mirror, with light entering from above and being absorbed by the structure. In Figures 4b,c, we use the Improvement to compare the performance of the Resonance-informed model and the Black-box model. For each sample in the test set, the Improvement is defined as follows: We first calculate the absorption spectrum of a sample using first principles methods, and then compute the absorption spectrum predicted by the Resonance-informed model. The mean absolute error (MAE) between the two spectra is calculated and denoted as M A E improved. Next, we replace the Resonance-informed model with the Black-box model and compute the MAE, denoted as M A E original. Improvement is defined as:

I m p r o v e m e n t = 1 − M A E i m p r o v e d M A E o r i g i n a l × 100 %

A positive Improvement indicates that, compared to the Black-box model, the absorption spectrum predicted by the Resonance-informed model is closer to the first principles result. We calculated the Improvement for each sample in the test set and generated fitting heatmaps with the structure parameters as coordinates, as shown in Figures 4b,c. The heatmaps predominantly show red regions, indicating that the Resonance-informed model generally outperforms the Black-box model. To further illustrate the differences between the two models, we present the simulation results for the first eight test samples in Figure 4d. It can be observed that for samples with a larger Improvement, the Resonance-informed model predicts the absorption spectra with significantly higher accuracy. In the resonance region, the Resonance-informed model identifies more resonances, and the heights of the resonance peaks are also more accurate. In the non-resonance region, the predictions of the Resonance-informed model are smoother and more closely aligned with the first principles results.

To investigate whether the performance improvement originates from the proposed resonance-informed decomposition or merely from increased model capacity, we conducted an ablation study on the 2D triangular texture set. As summarized in Table 1, we compared four models: the Black-box baseline, a Wider Black-box variant, a Fixed-split model, and our Resonance-informed model. The Wider Black-box variant scales the width of each layer in the baseline model by a factor of two, resulting in a model with approximately four times the parameters. This model serves as a direct control to isolate the effect of pure parameter count increase. The Fixed-split model employs two independent sub-networks that are rigidly assigned to predict the absorption spectra for the 800–950 nm and 950–1,100 nm wavelength bands, respectively. Except for the number of neurons in their final output layers (which matches the width of their assigned wavelength bands), the architectures of these two sub-networks are identical to the Black-box Model. This design approximately doubles the parameter count compared to the baseline. In contrast, our Resonance-informed model uses a learned Region Model to adaptively decompose the task into resonance and non-resonance regions, leading to a parameter count roughly three times that of the baseline.

TABLE 1. Ablation study on 2D triangular texture test set.

The bolded numbers indicate the best-performing set among the four models.

Table 1 shows that simply increasing model capacity is ineffective or even detrimental. The Wider Black-box model, despite having four times the parameters of the baseline, exhibits a 17.8% degradation in performance. Similarly, the Fixed-split model, with about twice the parameters, also performs slightly worse than the baseline. These results conclusively demonstrate that performance gains cannot be attributed to a mere increase in parameter count. In contrast, our Resonance-informed model achieves a 10.8% improvement over the baseline. The key distinction between it and the Fixed-split model is the use of the region model to dynamically divide the prediction regions for the two sub-models. This demonstrates that the region model has successfully captured meaningful structural or functional priors that are useful for neural network optimization, and that the performance improvement is a direct result of the proposed resonance-informed decomposition strategy, not model scaling.

We then altered the surface texture of the test structure by changing the shape from triangles to rectangles, while keeping all other simulation conditions unchanged. The variation ranges of L, b, h and H are from 600 to 800 nm, 100 to 200 nm, 100 to 200 nm, and 800 to 1,000 nm, respectively. As shown in Figures 5b,c, red areas again dominate the heatmaps, indicating that the Resonance-informed model continues to outperform the Black-box model in most cases, even after changing the surface texture. This suggests that the improvements brought by the Resonance-informed model are not limited to a specific texture shape.

Next, we increased H from the range of 800–1,000 nm to 3,000 to 3,200 nm. Other parameters remained the same with those in Figure 4. This modification aimed to test whether the Resonance-informed model continues to show superior performance compared to the Black-box model under more complex resonance conditions. The results are shown in Figures 6b,c. As in previous simulations, red areas dominate most of the heatmaps, indicating that the Resonance-informed model still performs better than the Black-box model in these more complex conditions.

In addition, we conducted simulations in three dimensions using a 3 dimensional circular matrix structure for light-trapping. The diagram of the 3 dimensional light-trapping structure is shown in Figure 7a. In the training samples, the cylindrical radius (r) ranged from 100 to 200 nm, and the height (h) ranged from 100 to 200 nm. L ranged from 600 to 800 nm, and H ranged from 3,000 to 3,200 nm. The heatmaps in Figure 7 show that red areas occupy most of the regions, indicating that, even in the most complex cases—such as three-dimensional and thickened structures—the Resonance-informed model outperforms the Black-box model. These simulations demonstrate that the Resonance-informed model provides a general improvement over the Black-box model across various light-trapping materials.

We also conducted simulations in the shorter wavelength region to assess whether our model performs well across a broader wavelength range, rather than being limited to specific wavelengths. The results are shown in Figure 8, where r ranges from 100 to 200 nm, and h ranges from 100 to 200 nm. L ranged from 600 to 800 nm, and H ranged from 800 to 1,000 nm. The heatmaps indicate that the Resonance-informed model significantly outperforms the Black-box model across the shorter wavelength range as well.

The proposed resonance-informed model demonstrates consistent and robust improvements over the black-box baseline across all five simulation configurations (Table 2). The model achieves lower MAE in every case. The superiority rate is the percentage of test samples where Resonance-informed model yields lower error than Black-box model. The superiority rate exceeds 70% in all simulations and reaches 95.75% for the thickened 2D triangular structure. This high consistency indicates that the performance gain is not merely an average effect but a reliable improvement applicable to the majority of samples. We note that the performance advantage is relatively smaller in the 3D cylindrical case. This may be attributed to the more intricate optical conditions and scattering pathways in three-dimensional geometries, which could make the initial region model less precise in delineating resonant and non-resonant domains. Further refinement of the region segmentation strategy for 3D systems presents a valuable direction for future work.

TABLE 2. Performance comparison between Black-box model and Resonance-informed model across different texture types.

The bolded numbers indicate the best-performing set among the four models.

Through the above simulations, we demonstrated the advantages of the Resonance-informed model over the Black-box model in predicting absorption spectrum. In the next section, we highlight the significant computational advantages of using neural network models over direct first principles calculations, with a speedup of at least three orders of magnitude. In Table 3, we show the time resources consumed by the Resonance-informed model and first principles calculations for absorption spectra with different number of sampling points (N). For the 2 dimensional (2D) rectangular structure, using the Resonance-informed model to predict the absorption spectrum results in a time reduction of about three to four orders of magnitude compared to first principles calculations. Moreover, we observed that as the number of sampling points increases, the time consumption for first principles calculations grows linearly, while the time consumption for the Resonance-informed model remains almost constant. This is because first principles calculations require repeated computations for each wavelength, while the Resonance-informed model can output the entire absorption spectrum simultaneously by adjusting the number of nodes in the output layer. Additionally, when moving from the 2 dimensional rectangular structure to the more complex 3 dimensional (3D) cylindrical structure, the time consumption for first principles calculations increases significantly, whereas the time consumption for the Resonance-informed model remains nearly constant. This indicates that, for the Resonance-informed model, the complexity of the physical system is effectively captured during the model’s training phase, which allows the prediction process to be significantly decoupled from the intricate details of the physical structure and thus greatly accelerated. Once the model is trained, the time consumption tends to remain consistent. In our simulations, the time consumption of the Black-box model was mostly under 1 ms, while the Resonance-informed model took about twice as long. This is expected, as the Resonance-informed model uses three sub-models, while the Black-box model has only one.

TABLE 3. Comparison of computational time for different absorption spectrum calculation methods under various conditions.

4 Discussion

In this study, we employ neural networks to separately predict the resonance and non-resonance regions of absorption spectra, with each region predicted independently. This approach significantly improves the accuracy of the original network. As demonstrated in the provided examples, both the resonance model and the non-resonance model exhibit limited performance when used in isolation. However, when combined, they produce more accurate predictions, indicating that each model is responsible for predicting different spectral regions. This approach has proven effective across a variety of cases.

We conducted simulations on four distinct materials with varying dimensions and surface textures to predict their absorption spectra. In a fifth simulation, we also varied the wavelength range of the predicted spectrum. The results consistently show that, across different material types, geometries, and wavelength ranges, the Resonance-informed model consistently outperforms the Black-box model. These findings highlight that integrating the phenomenological concept of resonance with statistical learning can universally enhance the performance of spectrum prediction.

Furthermore, by analyzing the performance of the two sub-models in the Resonance-informed model, we find that each sub-model specializes in predicting absorption spectra for either the resonance or non-resonance regions. This specialization not only improves the model’s accuracy but also enhances its interpretability, making the predictions more transparent and easier to understand.

5 Conclusion

In this paper, we propose a novel model for predicting absorption spectra by integrating the concept of resonance with machine learning neural networks. The new model significantly outperforms the baseline model across materials with varying shapes, dimensions, and spectral ranges. Moreover, the three sub-models of the proposed model yield intermediate results, which include predicting resonance probability, absorption spectra in the resonance region, and absorption spectra in the non-resonance region. These intermediate results enhance the interpretability of the model by shedding light on its prediction logic.

Our work demonstrates the potential of combining phenomenological theory with statistical learning, showing that this integration can improve both the performance and interpretability of neural networks. This approach is not limited to spectrum prediction; other phenomenological theories in physics can similarly assist statistical learning, helping models better align with human interpretation of physical phenomena, thereby enhancing their predictive accuracy and interpretability.

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Footnotes

Edited by: Lorenzo De Marinis, Sant’Anna School of Advanced Studies, Italy

Reviewed by: Tarik Akan, Bozok University, Türkiye

Tyler Huffman, U.S. Naval Research Laboratory, United States