introduction

As living standards have advanced, individuals’ lighting preferences have progressively transitioned from a primary focus on environmental sustainability and energy efficiency to an emphasis on health and comfort1,2. LED has become the prevailing choice for lighting applications due to their high efficiency, energy savings and long lifespan3,4,5. A blue semiconductor chip is typically employed as the primary light source in LED, with one or more phosphors used to convert part of the blue light into white light6.

The SPD is defined as the radiant power of a light source across different wavelengths. It is regarded as the “optical fingerprint” of the source. Key photometric and colorimetric parameters, including luminous flux, correlated color temperature (CCT), color rendering index (CRI), and circadian-related metrics, are obtained from the SPD.Consequently, accurate and controllable prediction of the SPD is particularly critical for engineering applications in healthy and customized lighting.

The SPD of an LED is governed by the coupled effects of several factors. These factors include drive current, junction temperature, phosphor type and mixing ratio, phosphor-to-silicone ratio, and secondary optical configuration. Variations in drive current and chip junction temperature can induce efficiency droop, band-gap shrinkage, and thermal quenching. As a result, the emission peak positions and the quantum efficiencies of the chip and the phosphors are altered7,8.

The relative spectral intensities in the mid- and long-wavelength regions are modulated by different phosphor ratios and energy conversion mechanisms. In the phosphor layer, absorption, scattering, and re-absorption are governed by the phosphor-to-silicone ratio and particle distribution. Spectral smoothness and angular color variation are thereby affected. Dynamic and accurate prediction of LED SPD is made highly challenging by the interactions among these factors within the LED system.

A number of studies have addressed SPD modeling and prediction. Fan et al. used artificial neural networks (ANNs) to develop an SPD decomposition and prediction method that correlates thermal effects under different combinations of drive current and case temperature with SPD9. However, this study did not explicitly address the coupled impacts of phosphor mixing ratio and phosphor-to-silicone ratio on SPD. From the perspective of physical modeling, J. C. C. Lo et al. used the excitation and emission spectra of multiple phosphors as inputs to predict the SPD of mixed systems, achieving high accuracy under known spectral priors10. In the work of Dupuis et al., the SPD was approximated using low-order polynomial functions, where the drive current was treated as the independent variable. A simple and easily deployed method was thus obtained. However, its capacity to represent complex coupling relationships was limited11.

From a statistical-modeling perspective, Gaussian, Lorentzian, and asymmetric double-sigmoid (Asym2sig) functions have been widely adopted to decompose and parameterize continuous LED spectra. Through such parametric representations, salient spectral features have been extracted, and the derived parameters have been further used to support downstream prediction tasks12,13,14,15. For example, M. H. Chang et al. combined multi-peak fitting with principal component analysis (PCA) for dimensionality reduction to enable rapid characterization and lifetime estimation16. Qian et al. modeled the SPD of phosphor-converted white LEDs (PC-wLEDs) using a superposition of two Asym2sig functions17. Fan et al. dynamically predicted CIE chromaticity coordinates, CCT, and CRI based on Gaussian and Lorentzian parameters for remaining-lifetime estimation18. Overall, extracting statistical feature parameters from SPD decomposition models has been demonstrated as an effective way to discretize continuous spectra while retaining key spectral characteristics.

In recent years, machine learning has been widely applied to data prediction in complex engineering problems19,20,21. Its core idea is to learn the mapping between inputs and outputs from data via parameter optimization and then perform effective inference on unseen samples. Among machine-learning approaches, artificial neural networks (ANNs) are regarded as one of the most popular supervised learning methods. They have been extensively used to handle diverse data types22,23. For instance, recent studies have demonstrated the powerful capability of ANNs in modeling complex non-linear systems, ranging from fault detection in power transmission networks24,25,26,27to performance prediction in optoelectronic devices. These successful applications indicate that data-driven models can effectively capture intrinsic physical relationships without relying on explicit analytical equations.

Applications to LED-related problems have been reported in several studies.For instance, Kaiyuan et al. used neural networks to predict LED lifetime under various currents and ambient temperatures. Yan, on the other hand, combined dynamic networks to enhance the robustness of LED lifetime prediction and reduce the prediction time28,29.

Research question

However, existing studies remain largely restricted to single-factor or limited-variable operating conditions. When confronted with complex scenarios involving highly coupled interactions among multiple phosphor ratios, phosphor-to-silicone ratios, and drive currents, targeted network models remain scarce. As a result, the development of new formulations and the implementation of complex processes still heavily rely on extensive trial-and-error experimentation. To address these challenges, a spectral prediction method is proposed. This method is based on Gaussian decomposition and an improved residual network. The masses of two phosphors, the phosphor-to-silicone ratio, and the drive current are used as inputs. Gaussian parameters are then outputted. The complete SPD is then reconstructed by Gaussian parameters. By leveraging the multi-peak characteristics inherent in LED spectra, this approach offers a parameterized representation of spectral shape endowed with clear physical interpretability. Concurrently, the neural network effectively models the nonlinear relationships arising from the interaction of multiple factors. Consequently, this framework facilitates rapid and precise SPD prediction for novel formulations across varying current levels without necessitating additional spectral measurements.

Contribution

The main contributions of this work can be summarized as follows:

  1. (1)

    A Gaussian mathematical model is established for SPD, providing a compact and physically interpretable parameterization. The model achieves \(R^{2} > 0.99\) for all samples.

  2. (2)

    An improved residual network with multi-head attention is proposed, specifically tailored for multi-factor LED systems. Compared with a conventional BP network and a baseline residual network, it significantly reduces RMSE, \(\Delta xy\), and the prediction error of the key amplitude parameter \(A_{1}\).

  3. (3)

    The proposed framework was validated using real packaged LED samples incorporating multiple phosphor formulations. Through these evaluations, the framework was shown to enable rapid spectrum prediction and to support customized spectrum design in multi-phosphor LED systems.

The remainder of this paper is organized as follows: “Theory and models” presents the SPD decomposition model and the neural network models. “Experimental setup and data acquisition” describes the test samples, experimental setup, and collected data used in this study. “Results and discussion” compares and discusses the prediction results obtained with different networks. Finally, “Conclusions” provides concluding remarks.

Theory and models

Gaussian mathematical model

The SPD of an LED is formed by the superposition of multiple emission peaks. These peaks are generated by the intrinsic emission of the blue chip and the stimulated emission of different phosphors. The overall spectral shape and colorimetric performance are determined by the relative intensities and positions of these peaks. Consequently, due to this multi-peak structure, a single function proves inadequate for describing the radiative properties, necessitating the use of multi-peak functional models. Gaussian functions are chosen for their simple form and high fitting flexibility. Peak positions, intensities, and widths of different emission components are effectively characterized. In this study, the Gaussian mathematical model is introduced to perform spectral decomposition and fitting of the SPD.The SPD at wavelength \(\lambda\) is expressed as

$$\begin{aligned} \operatorname {SPD}(\lambda )=\sum _{i=1}^{n} A_{i} \cdot \exp \left( -\frac{\left( \lambda -\lambda _{c, i}\right) ^{2}}{2 \sigma _{i}^{2}}\right) \end{aligned}$$
(1)

where \(A_{i}\) and \(\lambda _{c,i}\) denote the peak intensity and central wavelength of each emission component, respectively, and \(\sigma _{i}\) is the standard deviation that determines the peak width.

By summing multiple Gaussian components with appropriate weights, the full spectral curve can be reconstructed with high accuracy while preserving computational efficiency. This method can not only realize the fitting and decomposition of the spectrum, but also extract the parameters with physical significance for subsequent modeling and prediction.

Network models

The objective of this study is to achieve accurate prediction of LED SPD using machine-learning approaches. SPD prediction is reformulated by Gaussian parameters, rather than direct prediction of the full spectral curve. The phosphor mass ratios, the phosphor-to-silicone ratio, and the drive current are used as inputs.Gaussian parameters are output. The complete SPD is reconstructed by substituting the predicted parameters into the Gaussian mathematical model. Two major advantages are achieved. Physical interpretability is preserved. Each predicted parameter has clear spectral meaning(peak amplitude, central wavelength, or width). The output dimensionality is substantially reduced. The regression task is simplified. Predicting a small set of parameters is easier than predicting intensities at hundreds of wavelengths. The core of the model design is the construction of neural network architectures that capture the nonlinear mapping between the input feature vector and the Gaussian parameters under multi-factor coupling. Two network architectures are considered: a baseline residual network (Network I) and an improved residual network with multi-head attention (Network II). For comparison, a conventional BP neural network is also implemented.

Network I

Spectral prediction in multi-phosphor systems is characterized by pronounced nonlinearity and intricate variable interdependencies. Conventional backpropagation (BP) networks with shallow architectures are limited to capturing relatively simple nonlinear relationships. Moreover, increasing network depth often results in gradient vanishing or explosion, which compromises training stability and elevates the likelihood of convergence to suboptimal solutions. To alleviate these issues, a residual network was adopted. The architecture was deepened while training stability was preserved. Nonlinear fitting and feature representation were enhanced.

Fig. 1
Fig. 1
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The residual network structure.

The residual network structure is shown in Fig. 1. The input feature vector contains four elements: the masses of two phosphors, the phosphor-to-silicone ratio, and the drive current. These low-dimensional inputs were first projected into a higher-dimensional feature space through a fully connected layer with 128 units (Dense(128)). This layer is followed by batch normalization, a ReLU activation function, and dropout with a rate of 0.1, enabling nonlinear feature expansion while improving regularization. A second fully connected layer was subsequently applied to map the features into a 256-dimensional representation. This layer is also followed by batch normalization, a ReLU activation function, and dropout, with the dropout rate increased to 0.15. By expanding the feature capacity, this layer provides sufficient representational dimensionality for the residual learning blocks and enhances the network’s ability to model complex spectral characteristics. Deep feature extraction was then carried out through three sequentially stacked residual blocks.

Fig. 2
Fig. 2
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Internal structure of a residual block.

Each of these blocks is composed of a main pathway and a shortcut connection. As shown in Fig. 2, the output of a block is given by

$$\begin{aligned} y=\operatorname {ReLU}\left( F(x)+W_{s} x\right) \end{aligned}$$
(2)

In this formulation, F(x) denotes the residual mapping on the main path, which is implemented using two dense layers. Each layer is followed by a ReLU activation, batch normalization, and a dropout operation. The shortcut branch employs a linear projection matrix \(W_{s}\) so as to match the feature dimensions. For the first two blocks, the feature dimension is kept at 256, allowing high-capacity feature learning. In the third block, the number of units is reduced to 128. The projection matrix \(W_{s}\) used to map the 256 dimensions input to 128 dimensions, thereby enabling dimensionality reduction while preserving the residual connection. This design was structured to provide a large representational space in the intermediate layers, while the model complexity was constrained in the later stages. The residual connections were employed to mitigate gradient vanishing, preserve essential lower-level information, and enhanced the network’s nonlinear fitting capability. After the residual blocks, two additional fully connected layers were applied to further integrate the extracted features. Finally, the network outputed the Gaussian parameters. These parameters jointly define the Gaussian mathematical model of the spectrum and are used as the basis for spectral reconstruction and performance evaluation.

Network II

To further improve the accuracy of SPD prediction for LED, a multi-head attention module was introduced in Network I before the residual backbone. The improved residual network structure is shown in Fig.3. In this design, the intention was to better capture the complex interactions among phosphor ratios, the phosphor-to-silicone ratio, the drive current, and the resulting spectral shape.

Fig. 3
Fig. 3
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The improved residual network.

In this study, the input was treated as a fixed-length feature vector rather than a time-dependent sequence. To make use of attention without imposing an artificial temporal structure, the input features were arranged as a sequence of length one. In this way, subspace-level re-weighting can still be achieved, and the representation remains suitable for the static mapping from operating conditions to spectral parameters. In the attention module, the input feature vector was first processed through three independent linear projection layers. Through these layers, the vector was mapped into the query (Q), key (K), and value (V) representations. Each projection performs a 512\(\rightarrow\)512 transformation. These were then reshaped into eight attention heads with head dimension \(d_k\) = 64, yielding tensors of shape [B, 8, 1, 64], where B is the batch size. In each head, scaled dot-product attention is computed as

$$\begin{aligned} \text {scores} = \frac{QK^{\mathrm T}}{\sqrt{d_k}} \end{aligned}$$
(3)

followed by a softmax operation to obtain attention weights. After applying dropout to the weights, the weighted sum of V produced the output of each head. Because each operating factor affects the spectral shape in a different manner, the multi-head structure is used to learn separate weight patterns in different subspaces. Within this framework, certain heads are tasked with capturing the interaction between drive current and phosphor ratio, whereas others focus on the effects of scattering and re-absorption associated with the phosphor-to-silicone ratio. By leveraging these parallel representations, the model concurrently acquired multiple projections of the nonlinear interdependencies among factors. This approach is critical for enabling robust extrapolation across diverse formulations and effectively managing high-dimensional spectral fitting challenges. The outputs of all heads were concatenated back to a [B, 1, 512] tensor and fused via a final linear projection (512\(\rightarrow\)512). To improve the stability of training and to retain an explicit pathway for information flow, a post–layer-normalization residual scheme was applied. In this design, the output of the attention module was added to the original input through element-wise summation, and LayerNorm was performed afterward. The sequence dimension was then removed to form a [B, 512] tensor, which served as the input to a sequence of residual blocks for further deep feature extraction. Overfitting in small-sample regimes was suppressed through the combined use of residual connections, LayerNorm, and dropout.

Regarding the residual modules, both networks adopted a design in which a feed-forward main path was added to a residual path followed by a nonlinearity. However, Network II introduced slight modifications. Specifically, the main path in each block was designed with an expand-and-shrink procedure. The features were first projected into a higher-dimensional space and were then reduced back to the original size. Through this process, more informative interactions among intermediate representations could be formed. The residual branch was kept as an identity mapping, and the dimensionality of the block remained unchanged. Because the influences of drive current, phosphor ratio, and phosphor-to-silicone ratio on the spectral shape were strongly coupled and nonlinear, this strategy was adopted to improve representational capacity without causing unnecessary dimensional expansion.

In the training configuration and initialization approach, the settings employed in Network I were replaced by a more systematic framework in Network II. Network I relied on Keras with the Adam optimizer (learning rate set to \(1\times 10^{-3}\)), a ReduceLROnPlateau scheduler, and EarlyStopping under default parameter initialization. In contrast, Network II was implemented in PyTorch and was configured with the AdamW optimizer, where weight decay was explicitly applied to provide additional regularization. A StepLR schedule was introduced to control the learning rate in a piecewise manner. Gradient clipping was also incorporated, together with explicit Xavier initialization for linear layers and dedicated initialization for batch-normalization parameters.

Both networks were trained with mean square error (MSE) as the loss function, directly optimizing the Euclidean distance between predicted and true parameters. The predicted Gaussian parameters were then used in the Gaussian mathematical model to reconstruct the full SPD curve. The SPD is produced by the combined influence of blue-chip emission, phosphor conversion, scattering, and re-absorption. As a result, it naturally shows a structure that can be viewed as multi-channel generation and re-mixing. In this context, multi-head attention is used to perform dynamic re-weighting across channels and to divide the feature space into subspaces. This behavior is consistent with the way different physical channels contribute with different strengths during the optical process. In turn, it improves the adaptability of the model and its ability to generalize to unseen formulations. On this basis, residual blocks are applied to carry out deeper nonlinear integration. These two components therefore function in a complementary manner and jointly strengthen the representation of the input–output relationship. Consequently, the architecture becomes more appropriate for end-to-end regression of Gaussian parameters followed by spectral reconstruction. This section has systematically elaborated the architecture, core components, and design rationale of Network II. The next section will detail the data sources, preprocessing procedures, and experimental configurations.

Experimental setup and data acquisition

Experimental procedure

Two commercial phosphors were used in this study: a red phosphor LCN650 and a green phosphor LCG525F2. The encapsulant was a two-part silicone, used in mixed form. All materials were stored at room temperature and thoroughly mixed prior to use to ensure uniform dispersion.

The experiment consisted of two main stages. In the first stage, based on preliminary screening, a baseline formulation was established comprising 0.12 g of red phosphor, 1.56 g of green phosphor, and 21 g of mixed AB silicone. The corresponding red-to-green mass ratio was 1:13, and the phosphor-to-silicone ratio was 1:12.5. Preliminary optical characterization confirmed that this formulation yielded a CRI above 96 and a correlated color temperature between 3500 K and 4000 K, indicating good color performance. Subsequently, keeping the masses of red phosphor and silicone fixed, the mass of green phosphor was gradually increased in steps such that the red-to-green ratio changed by 0.5 units at each step. Samples with red-to-green ratios of 1:13.5, 1:14, 1:14.5, 1:15, 1:15.5, and 1:16 were prepared.

In the second stage, the baseline formulation was adjusted to 0.12 g of red phosphor, 1.26 g of green phosphor, and 15.87 g of silicone. The corresponding red-to-green mass ratio was 1:10.5, and the phosphor-to-silicone ratio was 1:11.5. Using the same incremental strategy, additional samples with red-to-green ratios of 1:11, 1:11.5, 1:12, 1:12.5, and 1:13 were fabricated.

To further investigate the influence of the phosphor-to-silicone ratio, additional comparative samples were prepared at the key red-to-green ratio of 1:13 by adjusting the silicone amount. Specifically, five samples with phosphor-to-silicone ratios of 1:9.4, 1:11.5, 1:12.5, 1:14.5, and 1:16.5 were fabricated to systematically analyze the effect of this ratio on optical performance.

All samples were prepared following standardized procedures. The phosphors were weighed using an analytical balance with a precision of 0.0001 g. The phosphors and silicone were mixed at room temperature for at least 3 minutes to achieve uniform dispersion. The mixtures were then immediately dispensed and cured to prevent performance degradation due to air exposure. In total, 16 samples with different formulations (covering variations in phosphor mass ratio and phosphor-to-silicone ratio) were prepared. One sample with a red-to-green ratio of 1:15.5 was partially damaged during fabrication and was therefore excluded from subsequent measurements to ensure data quality and experimental reliability.

The SPDs of the samples were measured with an integrated high-power LED spectroscopic testing system. This system is designed for rapid measurement and binning of optical, color, and electrical parameters of integrated high-power LED devices, as illustrated in Fig.4. It incorporates a fully programmable DC constant-current source, dedicated test fixtures, and an integrating sphere. All instruments were calibrated prior to measurement.

Fig. 4
Fig. 4
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Experimental setup.

To allow a comprehensive assessment under various drive conditions, eight current levels were included in the measurements: 150 mA, 200 mA, 250 mA, 300 mA, 350 mA, 400 mA, 450 mA, and 500 mA. The current levels were set from 150 to 500 mA in 50 mA increments and were chosen to cover the typical operating range of the LED.

Each sample was measured three times at every current level. This approach preserved the natural variations introduced by thermal effects and spectrometer noise. As a result, the training data were able to reflect the true output distribution of the physical system. In this way, the model was trained to learn the input–output mapping under realistic conditions and to maintain greater robustness against instrumental noise and system uncertainties. The recorded spectral measurement range is 380–780 nm, with a spectral resolution of 1 nm.

Data collection and preprocessing

Based on the experimental design, optical data were collected from 15 valid samples. This process yielded a total of 360 independent spectral samples. Each sample included the complete SPD together with its corresponding phosphor formulation parameters.

To evaluate the generalization ability of the models, a ratio-based stratified splitting method was used. Two formulations with red-to-green phosphor ratios of 1:12.5 and 1:14.5 were chosen as the test set. These ratios were not included in the training process, so they help to realistically assess how the model performs on new, unseen phosphor combinations. The other 13 formulations were used as the training set. This split ensures that the test set can properly evaluate prediction accuracy under unknown formulation conditions.

Because the number of training samples is limited and the network architecture is relatively deep, direct training may easily lead to overfitting. Therefore, a data augmentation strategy was adopted within the supervised learning framework. The motivation for the proposed data augmentation strategy stems from two physical considerations. First, experimental uncertainty modeling is performed by introducing Gaussian noise and parameter perturbations into both the input features and the output targets. This procedure is intended to simulate the inherent fluctuations arising from measurement errors and manufacturing tolerances in real experiments, thereby enhancing the robustness of the model to experimental noise. During the perturbation process, the intrinsic physical constraints of the input parameters are strictly preserved.

Second, sample enrichment based on physical continuity is achieved by generating additional samples through convex combinations of valid experimental data. This approach effectively increases the sampling density in the high-dimensional formulation space, enabling the network to learn smooth transition patterns between sparsely distributed experimental data points. Data augmentation was performed exclusively on the training set. The test set was kept completely separate from any augmentation or statistical estimation processes to prevent information leakage. With this enlarged dataset, the neural network received more diverse training examples, which helped improve its learning performance.

All experiments were conducted under controlled and identical ambient conditions, with the room temperature maintained at approximately \(25^{\circ }\)C. Under these conditions, variations in junction temperature are primarily correlated with the driving current, which is explicitly included as an input variable in the model. Therefore, junction temperature was not treated as an independent variable in this study. Instead, it was implicitly accounted for through the current-dependent behavior. This modeling choice allows the present work to focus on the effects of the core variables–phosphor mixing ratio, phosphor-to-binder ratio, and driving current—on the spectral power distribution.

Throughout the experimental campaign, a stringent quality-control protocol was implemented. Measurement instruments were periodically recalibrated to maintain accuracy and consistency. Fabrication parameters such as mixing duration, ambient temperature, and curing conditions were carefully documented. These efforts provided a high-quality data foundation for subsequent neural network modeling and spectral reconstruction.

Results and discussion

Gaussian decomposition

Based on the SPD characteristics of the LED used in this study, three Gaussian components were determined to be an optimal choice for accurate yet compact spectral reconstruction.

The Fig. 5 shows the SPD reconstructed using Gaussian fitting. Four representative samples were selected to visually demonstrate the spectral fitting results of the Gaussian mathematical model. It shows the original SPD, the three extracted spectral components, and the combined peak model formed by adding these Gaussian components together. A total of 64 independent samples were considered to validate the Gaussian mathematical model, selected from optical data obtained under eight current levels for the 15 valid samples.

Fig. 5
Fig. 5
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SPD decomposition modeling with the Gaussian function fitting.

The coefficient of determination (\(R^2\)) was used to evaluate the fitting quality. \(R^2\) is defined as

$$\begin{aligned} R^2 = 1 - \frac{\sum _{j=1}^{n} \left( I_m(\lambda _j) - I_e(\lambda _j) \right) ^2}{\sum _{j=1}^{n} \left( I_m(\lambda _j) - \bar{I}_m \right) ^2} \end{aligned}$$
(4)

The measured spectral power density at wavelength \(\lambda _j\) is denoted by \(I_m(\lambda _j)\) , and the reconstructed value is denoted by \(I_e(\lambda _j)\). The mean of the measured SPD across the wavelength range is represented by \(\bar{I}_m\). The number of wavelength sampling points is denoted by n . In this study, the wavelength range was set from 380 nm to 780 nm with a step size of 1 nm. This configuration produced a total of n = 401 points.

The value of \(R^2\) lies between 0 and 1, with values closer to 1 indicating better agreement between the reconstructed curve and the measured data. As shown in Table1, all \(R^2\) values are larger than 0.99. These results demonstrate that the original LED spectra can be accurately represented using the Gaussian-based SPD decomposition method.

Table 1 Goodness-of-fit results (\(R^2\)).
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Prediction with Network I

The collected spectral samples were then used for training and prediction. After data augmentation, 2000 spectral samples were used as the training set, whereas the 48 pre-assigned spectral samples were reserved as the test set. The test set covers different formulations and drive currents and thus provides a meaningful evaluation of generalization performance.

Network I was trained to convergence and then applied to the test set to obtain predicted Gaussian parameters. To quantify prediction accuracy, the absolute percentage error of each output parameter was computed as

$$\begin{aligned} E_p = \frac{|P - R|}{|R|} \times 100\% \end{aligned}$$
(5)

where \(E_p\) is the percentage error, R is the true Gaussian parameter, and P is the predicted parameter.

Figure 6 shows the absolute percentage of prediction error. The parameters exhibit different error levels, which are closely related to their physical roles in constructing the spectrum. The amplitude parameters (\(A_{1}-A_{3}\)) represent the intensities of the emission peaks. These values are mainly affected by phosphor absorption efficiency, energy-transfer behavior, and the phosphor-to-silicone ratio. Among them, the mean error of \(A_{1}\) is significantly higher than those of the other parameters, reaching 6.894%, with a maximum error close to 15%. This suggests that the baseline residual network has limited capability in capturing the energy conversion mechanisms between the blue chip and phosphors, particularly in accurately fitting the short-wavelength emission intensity. In contrast, the errors of \(A_{2}\) and \(A_{3}\) are clearly lower, indicating that the model more reliably predicts emission intensities in the mid- and long-wavelength regions.

Fig. 6
Fig. 6
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Prediction errors of characteristic parameters in Gaussian model with Network I.

The center-wavelength parameters (\(\lambda _{c, 1}-\lambda _{c, 3}\)) were predicted with the highest level of accuracy. Their mean errors are kept below 0.43%, and the maximum deviations are substantially lower than those observed for the amplitude parameters. This outcome is consistent with the physical behavior of the system. The central wavelengths are mainly governed by the intrinsic energy-level structure of the phosphors and by the surrounding lattice environment. Because these energy levels remain relatively stable and show limited sensitivity to external operating conditions, their variations were captured by the model with high reliability.

The standard deviation parameters (\(\sigma _{1}-\sigma _{3}\)) were predicted with error levels that fall between those of the amplitude and center-wavelength parameters. Their mean errors remain below 2.0%. The maximum error of \(\sigma _{1}\) reaches 6.46%. This value is considered reasonable because the peak width is influenced by lattice defects, non-radiative relaxation pathways, and phonon interactions. These physical processes can introduce uncertainties under different measurement conditions. Overall, the prediction errors associated with the standard deviation parameters stay within an acceptable range, indicating that the model was able to represent the mechanisms responsible for spectral broadening.

In summary, the prediction errors associated with the amplitude parameters are observed to be greater than those related to the center-wavelength and standard-deviation parameters. This trend can be attributed to the fact that peak intensities are influenced by a broader range of factors. Specifically, the amplitude is governed not only by the intrinsic luminescence efficiency but also by variables such as doping concentration, energy transfer efficiency, and the phosphor-to-silicone ratio. The intricate and nonlinear interactions among these variables pose significant challenges for the model to capture with complete accuracy. In contrast, the center wavelengths demonstrate greater physical stability, and the peak widths exhibit only moderate sensitivity to external conditions. Consequently, these parameters were predicted with comparatively higher precision.

Although single-parameter predictions are informative, the perceived SPD is determined by the joint action of all parameters. Therefore, the three Gaussian curves reconstructed from the predicted parameters were superimposed and compared with the measured spectra to evaluate prediction reliability at the spectral level.

For quantitative evaluation of spectral reconstruction, two metrics were used: Root-Mean-Square-Error (RMSE) and CIE 1931 chromaticity difference (\(\Delta xy\)). RMSE measures the amplitude deviation between reconstructed and measured spectra over the wavelength domain. \(\Delta xy\) quantifies the deviation of the predicted chromaticity from the reference. Those can be expressed as Eqs. 6 and 7.

$$\begin{aligned} \textrm{RMSE} = \sqrt{\frac{1}{n}\sum _{j=1}^{n}\left( I_m(\lambda _j)-I_e(\lambda _j)\right) ^2} \end{aligned}$$
(6)

where \(I_m(\lambda _j)\) is real measured SPD, \(I_e(\lambda _j)\) is predicted value of SPD, and n is number of measurements.

$$\begin{aligned} \Delta xy = \sqrt{(x_m - x_e)^2 + (y_m - y_e)^2} \end{aligned}$$
(7)

where \(x_m\) , \(y_m\) are the CIE 1931 xy coordinates computed from the reconstructed SPD and \(x_e\), \(y_e\) are the expected coordinates from the measured SPD.

Table 2 Comparison of networks on the test set.
Full size table

As shown in Table.2, the mean RMSE of Network I on the test set is 2.937, indicating moderate numerical discrepancies between the predicted and measured spectra. The mean \(\Delta xy\) is 0.008032, suggesting small chromaticity shifts and good color consistency. To more intuitively demonstrate SPD prediction performance, four representative test samples were selected, and their measured SPD, Gaussian-fitted SPD, and neural-network-predicted SPD were plotted.

Fig. 7
Fig. 7
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SPD prediction results with Network I.

As depicted in Fig. 7, the predicted SPD curves closely match the measured spectra, confirming the feasibility of the residual network for reconstructing spectral characteristics.

Prediction with Network II

The prediction results of the characteristic parameters for Network II are illustrated in Fig. 8.

Fig. 8
Fig. 8
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Prediction errors of characteristic parameters in Gaussian model with Network II.

It can be seen that the maximum error of \(A_{1}\) is reduced from 14.713% to 7.172%. Its mean error is also lowered from 6.894% to 2.165%. The prediction errors of \(A_{2}\) and \(A_{3}\) remain lower than those of \(A_{1}\). This pattern suggests that the model is able to learn the energy-conversion behavior between the blue chip and the phosphors. This improvement is mainly due to the attention mechanism, which helps the network focus on important features, and to the residual structure, which supports the learning of complex nonlinear relations. The mean errors of the center-wavelength parameters are all kept below 0.65%.They continue to exhibit the highest prediction accuracy, consistent with their underlying physical properties. Moreover, the maximum error of \(\sigma _{1}\) is reduced from 6.894% to 4.353%. The mean errors of the standard deviation parameters remain comparable to those produced by Network I. Overall, the peak-broadening characteristics are represented in a stable and reliable manner by the model. To further illustrate SPD prediction, four sets of conditions shown in Fig.9 were used to compare the measured SPD, the Gaussian-fitted model, and the predicted SPD.

Fig. 9
Fig. 9
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SPD prediction results with Network II.

Similarly, RMSE and \(\Delta xy\) were calculated, and as shown in Table 2. The mean RMSE decreases from 2.937 to 2.556, while the mean \(\Delta xy\) decreases from 0.008032 to 0.004157. This indicates that the improved model provides a clear performance enhancement. In LED spectral prediction studies, BP networks are among the most widely used models, so a BP network was constructed and evaluated on the same training and test datasets for comparison.The BP network is built with a classical three-layer feed-forward structure that includes an input layer, a hidden layer, and an output layer. The hidden layer is activated by ReLU, and the output layer applies a linear function to match the requirements of a regression task. The mean-squared error is used as the loss function, and standard gradient descent is adopted as the optimizer.

As shown in Fig. 10, the conventional BP network performs worse in parameter prediction. When compared with Network II, the mean error of \(A_{1}\) increases notably from 2.165% to 9.298%, and its maximum error rises from 7.172% to 19.946%. The mean errors of \(A_{2}\) and \(A_{3}\) also increase, changing from 1.658% and 1.489% to 3.417% and 2.206%, respectively. Other parameters still keep relatively low error values, but the overall trend indicates weaker predictive ability. As shown in table2, in terms of overall performance, the BP network yields a mean RMSE of 3.461015 and a mean \(\Delta xy\) of 0.009392. These values are 35.41% and 125.93% higher than those achieved by the Network II. This result indicates that the conventional BP network has limited capability in modeling complex energy coupling and nonlinear spectral mapping.

Fig. 10
Fig. 10
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Prediction errors of characteristic parameters in Gaussian model with BP-NN.

In summary, the performance ranking of the three network models on the test dataset is as follows. The improved residual network (Network II) achieves the best performance, followed by the baseline residual network (Network I), while the BP neural network performs the worst.

The conventional BP network produces relatively large prediction errors in multi-phosphor systems. It does not capture the energy-conversion behavior in the short-wavelength region effectively. In contrast, the residual network improves gradient propagation through its deeper structure and the use of skip connections. As a result, its overall prediction accuracy is increased. Building on this structure, the proposed improved residual network further introduces attention mechanisms. This addition strengthens the extraction and fusion of spectral features, resulting in superior prediction accuracy and generalization ability.

The comparative results show that the improved network presented in this work performs markedly better than the other models in terms of RMSE, \(\Delta xy\), and amplitude-parameter prediction accuracy. These findings demonstrate its strong generalization ability and modeling advantages in multi-phosphor systems.

Significance analysis

To rigorously evaluate whether the performance differences among the Network II, Network I, and BP network are statistically significant, non-parametric statistical tests were conducted. Since the error distributions do not strictly follow a normal distribution, the Friedman test was first employed to examine the overall differences among the three models. After rejecting the null hypothesis, the Wilcoxon signed-rank test30,31 was further applied for pairwise comparisons to identify specific performance differences. The statistical analysis results are summarized in Table 3.

Table 3 Statistical significance analysis of model comparisons.
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The Friedman test revealed significant differences among the three models for all evaluation metrics, including RMSE (\(p = 3.33 \times 10^{-9}\)), \(\Delta xy\) (\(p = 1.46 \times 10^{-10}\)), and \(A_{1}\) error (\(p = 1.12 \times 10^{-12}\)). Further pairwise comparisons using the Wilcoxon signed-rank test demonstrate that the proposed Network II significantly outperforms both the Baseline Network I and the BP network across all metrics (p < 0.001). These results statistically confirm that the observed performance improvements are robust and not due to random chance.

Multi-model comparative analysis

To comprehensively evaluate the performance of the proposed model, we expanded the scope of the comparative analysis by incorporating four representative machine learning models that have been widely adopted in recent literature. Specifically, support vector machine (SVM), Gaussian process regression (GPR), decision tree32 (DT), and random forest (RF) models were selected for comparison.

Table 4 Multi-model comparison results.
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As shown in Table 4, Network II and SVM exhibit comparable performance in standard regression metrics. However, regarding the \(\Delta xy\), which measures color fidelity, Network II outperforms SVM (0.0042 vs. 0.0054). This indicates that the proposed model better preserves spectral characteristics critical for accurate color reproduction. Other models (RF, GPR, DT) show significantly larger errors. Observed performance gaps can be attributed to the following aspects. Specifically, SVM typically treats multi-output regression as independent tasks, ignoring correlations between spectral parameters. Tree-based models (RF and DT) produce piecewise-constant predictions, which are ill-suited for smooth spectral reconstruction. GPR struggles with kernel generalization in high-dimensional mapping tasks. In contrast, the proposed Network II effectively learns continuous, non-linear mappings and parameter correlations, ensuring more robust and accurate spectral predictions.

Ablation experiment

To assess the effectiveness of the multi-head attention module in our proposed Network II, we conducted an ablation experiment. We compared the performance of the full model against a simplified version where the attention module was removed ( Network II w/o ), while keeping all other hyperparameters unchanged. The comparison results are presented in Table 5.

Table 5 Ablation study results of Network II.
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The results indicate that the full model outperforms the ablated version across evaluation metrics. Specifically, the RMSE decreased from 2.6135 to 2.5558, and the MAE decreased from 2.04 to 1.9904. More importantly, the ablation study reveals the specific contribution of the attention mechanism to parameter estimation. As observed in the last column of Table 5, the removal of the attention module led to a significant increase in the mean error of parameter \(A_{1}\) , rising from 2.165 to 4.043. \(A_{1}\) is mainly affected by phosphor absorption efficiency and the phosphor-to-silicone ratio. This result demonstrates that the attention mechanism plays a critical role in modeling the energy transfer process between the blue chip and the phosphors. In conclusion, the attention module is essential for the proposed architecture, particularly for enhancing the model’s sensitivity to key parameters like \(A_{1}\) without compromising the overall stability.

Robustness evaluation

To further assess the robustness of the proposed model, we conducted a sensitivity analysis by introducing controlled perturbations into the input parameters of the test set. Specifically, we introduced Gaussian noise with magnitudes of 1%, 5%, and 10% respectively into the input features of the test set.

Table 6 Sensitivity analysis results under different perturbation levels.
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The sensitivity analysis results are reported in Table 6. As the perturbation level increases, the prediction errors exhibit a gradual and monotonic increase across all metrics. Notably, even under a 10% perturbation, the performance degradation remains limited, and no abrupt error escalation is observed. This indicates that the proposed Network II learns a smooth and stable mapping between the input parameters and spectral characteristics, demonstrating strong robustness against input uncertainty and measurement noise. Such robustness is particularly desirable for practical LED applications, where measurement noise and manufacturing tolerances are unavoidable.

Generalization evaluation

To further validate the generalization capability and robustness of the proposed model, we conducted an additional evaluation on a completely independent dataset collected under experimental conditions different from the training phase. The independent test set consists of 36 samples designed to test the model’s performance on both unseen material formulations and unseen operating conditions. The dataset includes samples with red-to-green phosphor ratios of 1:12.5 and 1:14.5, which represent ”unseen” formulations not present in the training data. In addition, we further extended the test set by incorporating samples with phosphor ratios of 1:11 and 1:16. All samples were measured under driving currents of 100 mA, 325 mA, and 550 mA . Likewise, triplicate measurements at each current level preserved the natural variations induced by thermal effects and spectrometer noise. Crucially, these specific current levels were never used during the model training process, thereby testing the model’s ability to generalize to new measurement environments.

Table 7 Generalization performance on independent test datasets.
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The prediction results on the extended test set are summarized in Table 7. Despite the fact that the test samples include unseen phosphor formulations and previously unobserved driving currents, the proposed model achieves an average RMSE of 2.34 and an MAE of 1.82, which are comparable to the results obtained on the original test set. This indicates that the model maintains stable predictive performance under distribution shifts in both material composition and operating conditions. Meanwhile, good generalization performance is observed for both unseen phosphor ratios and previously unobserved driving currents. This result confirms the effectiveness of the physically informed Gaussian parameterization and the improved residual network in capturing the underlying spectral response mechanisms.

Strengths, limitations, and future directions

In the context of growing emphasis on healthy lighting, accurate spectral prediction is essential for optimizing circadian effectiveness, visual comfort, and color fidelity. Traditional methods depend heavily on time-consuming trial-and-error experiments. In contrast, the proposed prediction framework offers a highly efficient alternative. With a lightweight architecture of approximately 4.39 million parameters, the model achieves an average inference latency of just 4.47 ms on a standard CPU, enabling real-time spectral estimation without specialized high-performance computing infrastructure. This physics-informed approach, rooted in rigorous experimental data and interpretable Gaussian parameterization, ensures high data fidelity and transparency. Consequently, LED manufacturers are provided with a practical, reliable mapping tool that links material composition and drive current to the resulting spectrum. By integrating this decision-support tool into existing workflows, product development cycles can be significantly shortened, and R&D costs reduced, effectively bridging the gap between theoretical modeling and industrial mass production.

Despite the promising results, this study has several limitations. First, the validation is currently restricted to a dual-phosphor LED system; the framework’s scalability to complex multi-component systems remains to be verified. In addition, factors such as junction temperature and long-term aging were not explicitly modeled, as the focus was on the dominant coupling effects of composition and current. Finally, although data augmentation was employed, the dataset scale is limited compared to industrial standards.

Future work will aim to broaden the phosphor systems and phosphor-to-silicone ratios. Furthermore, work will also be directed toward introducing multi-task learning so that SPD, CRI, luminous efficacy, and other performance indicators can be predicted together. These extensions are expected to offer more comprehensive evaluations and to support the intelligent optimization of next-generation healthy lighting and high–color-quality products.

Conclusions

In this study, a hybrid SPD prediction framework was developed by integrating a Gaussian mathematical model with an improved residual neural network. The Gaussian model achieved high-fidelity spectral representation (\(R^2\) > 0.99), enabling the network to effectively map phosphor composition and drive current to spectral parameters. Comparative analysis demonstrated that the improved architecture, incorporating multi-head attention, outperformed the baseline. Specifically, the mean error of amplitude \(A_{1}\) was reduced from 8.262% to 2.165%, spectral reconstruction RMSE was decreased from 2.937 to 2.556, and chromaticity deviation (\(\Delta xy\)) was lowered from 0.008032 to 0.004157. These results confirm that the proposed model provides a precise and computationally efficient tool for the rapid development of customized lighting.

Despite these achievements, limitations exist. The framework is currently validated on a dual-phosphor system, and factors such as junction temperature and long-term aging were not explicitly modeled. Additionally, the dataset scale is limited compared to industrial standards. Future work will focus on expanding to broader phosphor systems and implementing multi-task learning to predict SPD alongside CRI and luminous efficacy.