4.1. Lookup block: Global parameter-to-spectrum mapping

The first stage of SM-Net is the Lookup Block, which transforms the three-dimensional input parameters into a coarse, low-frequency spectral prediction. In terms of the architecture in Figure 5, this stage consists of:

• • a Parameter Embedding Layer that expands the physical inputs into a higher-dimensional representation,

• • three Fully Connected Layers, with dimensions $3 \rightarrow 4\,096 \rightarrow 16\,384 \rightarrow 11\,655$ , implemented using residual MLP blocks,

• • a Coarse Spectrum Generator that produces an initial spectral estimate.

The embedded parameters pass through two residual fully connected layers:

(1) \begin{align}\mathbf{h}_1 &= f_1(\mathbf{x}) + P_1(\mathbf{x}), \end{align}

(2) \begin{align}\mathbf{h}_2 &= f_2(\mathbf{h}_1) + P_2(\mathbf{x}), \end{align}

where $f_1$ and $f_2$ are residual MLP blocks, and $P_1, P_2$ are linear skip projections that forward the physical parameters directly to deeper layers.

The Coarse Spectrum Generator produces the initial coarse spectrum:

(3) \begin{equation}\mathbf{s}_{\mathrm{coarse}} = W_3 \mathbf{h}_2 + P_3(\mathbf{x}),\end{equation}

where $W_3$ is a learned linear operator and $P_3$ is the final linear layer of this block, mapping output to the spectral dimension.

A learnable per-wavelength baseline vector $\mathbf{b} \in \mathbb{R}^{L}$ is initialised to the median flux of the training data. This baseline acts as a stabilising prior on the predicted spectrum. It is applied using a Softplus transformation,

(4) \begin{equation}\mathbf{s}_0 = \mathrm{Softplus}_\beta\!\left( \mathbf{s}_{\mathrm{coarse}} + \mathbf{b} \right)\!,\end{equation}

which guarantees non-negative output fluxes. The parameter $\beta$ controls the sharpness of the Softplus activation.

4.2. Refinement block: Residual spectral refinement

The second stage of SM-Net is the Refinement Block, which applies a series of convolutional operations to refine the coarse prediction with detailed wavelength-dependent corrections. Architecturally, this block consists of:

• • an Input Conditioning Layer that integrates the learned conditioning vector, a fixed three-channel positional encoding, and the coarse spectrum;

• • a Multi-scale Feature Extractor consisting of three parallel convolutional branches with kernel sizes of 7, 15, and 23;

• • a Fusion Block that aggregates features from the parallel branches;

• • a Refinement Output Layer that predicts the additive spectral correction.

The stellar parameters are first expanded along the wavelength axis by a learned linear map:

(5) \begin{equation} \mathbf{c} = Q\,\mathbf{x} + \mathbf{b}, \end{equation}

where

• • $\mathbf{x} \in \mathbb{R}^{D}$ is the input parameter vector (where $D = 3$ );

• • $Q \in \mathbb{R}^{L \times D}$ is the learned weight matrix projecting parameters onto the spectral axis;

• • $\mathbf{b} \in \mathbb{R}^{L}$ is the bias vector;

• • $\mathbf{c} \in \mathbb{R}^{L}$ is the resulting conditioning vector.

Here L denotes the length of the spectrum.

To provide the refinement module with an explicit notion of spectral position, we generate a fixed three-channel positional encoding $\mathbf{P} \in \mathbb{R}^{3 \times L}$ . For a normalised wavelength index $t_i \in [0, 1]$ at bin i, the encoding channels are defined as:

(6) \begin{equation} \mathbf{P}_i = \begin{bmatrix} t_i \\ \sin\! (2\pi t_i) \\ \cos\!(2\pi t_i) \end{bmatrix}. \end{equation}

The Input Conditioning Layer concatenates the coarse spectrum $\mathbf{s}_0$ , the conditioning vector $\mathbf{c}$ , and the three-channel positional encoding $\mathbf{P}$ along the channel dimension:

(7) \begin{equation} \mathbf{X} = \begin{bmatrix} \mathbf{s}_0^{\top} \\ \mathbf{c}^{\top} \\ \mathbf{P} \end{bmatrix} \in \mathbb{R}^{5 \times L}. \end{equation}

Thus, $\mathbf{X}$ contains one channel from $\mathbf{s}_0$ , one channel from $\mathbf{c}$ , and three channels from $\mathbf{P}$ .

The Multi-scale Feature Extractor processes $\mathbf{X}$ through three parallel branches to capture features at different scales. For each kernel size $k \in \{7, 15, 23\}$ , the output $\mathbf{u}_k$ is computed as:

(8) \begin{equation} \mathbf{u}_k = \mathcal{D}\left( \phi\left( \mathrm{Conv}_k\left( \mathrm{pad}_k(\mathbf{X}) \right) \right) \right)\!, \end{equation}

where $\mathrm{pad}_k(\!\cdot\!)$ denotes replicate padding, $\phi$ is the GELU activation, and $\mathcal{D}$ denotes Dropout. The feature maps from the parallel branches, each corresponding to a different kernel size k, are concatenated along the channel dimension to form a composite multi-scale feature map $\mathbf{U}$ . This aggregation allows the network to process local spectral lines and broader continuum features simultaneously. The composite map is processed by a Fusion Block:

(9) \begin{equation} \mathbf{h} = \mathcal{D}\left( \phi\left( \mathrm{Conv}_{\mathrm{fuse}}(\mathbf{U}) \right) \right)\!. \end{equation}

Here, $\mathrm{Conv}_{\mathrm{fuse}}$ represents a convolutional layer designed to mix information across the spectral channels. Finally, the Refinement Output Layer projects these fused features down to a single dimension to produce the scalar spectral correction $\Delta\mathbf{s}$ :

(10) \begin{equation} \Delta\mathbf{s} = \mathrm{Conv}_{\mathrm{out}}(\mathbf{h}). \end{equation}

The final high-resolution spectrum is obtained via a residual connection, where the learned correction is added to the initial coarse estimate:

(11) \begin{equation} \mathbf{s}_{\mathrm{final}} = \mathbf{s}_0 + \Delta\mathbf{s}. \end{equation}

4.3. Loss function and treatment of masked flux

To handle masked or zero-valued flux (originating from incomplete line lists or numerical issues), SM-Net uses a weighted mean-squared error (MSE) loss:

(12) \begin{equation}\mathcal{L} =\frac{1}{BL}\sum_{i=1}^{B}\sum_{\lambda=1}^{L}w_\lambda\, m_{i,\lambda}\,\left( s_{i,\lambda} - t_{i,\lambda} \right)^2,\end{equation}

where:

• • B is the batch size,

• • $w_\lambda$ is a per-wavelength weight,

• • $t_{i,\lambda}$ and $s_{i,\lambda}$ are the target and predicted fluxes,

• • $m_{i,\lambda}$ is a binary mask:

  \begin{align*}m_{i,\lambda} =\begin{cases}0, & t_{i,\lambda} = 0 \ (\text{masked or uncertain flux}),\\1, & \text{otherwise}.\end{cases}\end{align*}

Masked flux values therefore exert no penalty, allowing the model to reconstruct missing structure by relying on neighbouring wavelengths and on the broader learned relationships encoded in the network.

4.4. Training strategy

The SM-Net model was first trained, evaluated and refined separately on each of the four core libraries. This initial evaluation phase not only guided model optimisation but also revealed regions within individual libraries where the spectra exhibited internal inconsistencies. Below 3 000 Å individual-library training and direct cross-library inspection revealed abrupt continuum offsets, discontinuous slopes, and mutually inconsistent flux amplitudes at neighbouring or matched parameter locations; this region was therefore excluded rather than treated as a physically coherent common domain.

Model training used the Adam optimiser for Learning Rate (LR) scheduling, with an initial LR of $4\times 10^{-4}$ and an initial batch size of $1\,024$ . Whenever the validation loss failed to improve for ten consecutive epochs, both the LR and batch size were reduced by a factor of $0.8$ , with a minimum batch size capped at 128. Early stopping was triggered after 50 epochs without improvement. This decay schedule in LR and reduction in training batch size, shown as red dotted lines in Figure 6, served two complementary roles. Large batches in early epochs stabilised the optimisation landscape and enabled the network to learn broad, global spectral trends, whereas progressively smaller batches introduced beneficial gradient noise that helped the optimiser escape shallow minima and focus on fine-grained spectral structure. Reducing the LR in tandem ensured that later stages of training refined subtle features without overshooting, which is essential for high-resolution spectra, where neighbouring wavelengths differ only marginally in flux. Empirically, this coupled decay of batch size and LR yielded smoother convergence than adjusting either hyper-parameter alone.

Figure 6.

Training and validation losses throughout the training process. Red lines indicate the epochs at which the learning rate and the batch size were reduced, both by a factor of 0.8.

We enabled automatic mixed-precision with CUDA, using bfloat16 throughout training without gradient scaling. This reduced memory pressure during convolutional refinement without degrading accuracy. Training required a peak GPU memory of 21 GB, whereas inference with a batch size of 256 spectra requires about $4.5$ GB of GPU memory. The model achieves rates of approximately $14\,000$ spectra per second on a single NVIDIA RTX 4090 GPU. For maximum throughput and full GPU utilisation, the number of inferred spectra should exceed a few thousand. For small inference jobs, the GPU is under-utilised, kernel-launch overheads dominate, and the resulting throughput is noticeably lower than the peak rate.

Figure 7.

Mean and median residuals between matched PHOENIX–Husser and C3K spectra as a function of $\log_{10}\kern-1.5pt(\lambda/$ Å). Residuals are defined as Husser $-$ C3K and are computed in the log1p-scaled flux representation across 783 matched $(T_{\text{eff}}, \log g, \log Z)$ grid points. The shaded region indicates the $\pm 1\sigma$ dispersion of the residuals.

Figure 8.

Mean squared difference between matched PHOENIX–Husser and C3K spectra across the common parameter grid, shown in the $\log_{10}\kern-1.5pt(T_{\text{eff}}/\text{K})$ – $\log g$ plane. The vertical axis shows $\log g$ . Values are computed in the log1p-scaled flux representation and averaged over $\log Z$ .

6.1. Error distributions across the parameter space

To diagnose potential biases on the non-uniform combined grid, Figures 10–12 show the MSE distribution as a function of $\log_{10}\kern-1.5pt(T_{\mathrm{eff}}/\mathrm{K})$ , $\log g$ , and $\log Z$ parameters across all splits, respectively. One can draw the following conclusions based on these MSE box plots.

The train, validation, and test splits exhibit very similar MSE profiles as functions of $T_{\mathrm{eff}}$ , $\log g$ , and $\log Z$ . Although the absolute values differ slightly, the overall shapes of the distributions and the locations of the outliers are consistent across all three splits. Together with the close agreement in the summary statistics in Table 4, this indicates that SM-Net does not overfit and generalises robustly across unseen regions of the parameter grid.

The box plots in Figures 13–15 show the distribution of MSE values for the test set, stratified by stellar parameter and separated by the four constituent libraries in the combined dataset. For TMAP, the reconstruction errors are systematically smaller than those of the other libraries. This is likely because TMAP occupies a region of parameter space where the spectra are comparatively smooth and contain fewer strong, rapidly varying features, while also having minimal overlap with the other libraries and therefore fewer cross-library inconsistencies. Figure 15 shows the corresponding $\log Z$ distributions only for PHOENIX–Husser and C3K, since OB–PoWR and TMAP do not include metallicity as an explicit physical parameter in their source grids.

Figure 13.

Distribution of MSE across $\log_{10}\kern-1.5pt(T_{\text{eff}}/\text{K})$ for the four libraries.

Figure 13. Long description

Panel A: Box plot for PHOENIX-Husser. The x-axis is labeled log10(Teff/K) ranging from 3.30 to 4.90, and the y-axis is labeled MSE (log1p flux) ranging from 10^-6 to 10^-3. The plot shows median values and outliers across different effective temperatures. Panel B: Box plot for C3K. The x-axis is labeled log10(Teff/K) ranging from 3.30 to 4.90, and the y-axis is labeled MSE (log1p flux) ranging from 10^-6 to 10^-3. The plot shows median values and outliers across different effective temperatures. Panel C: Box plot for OB–PoWR. The x-axis is labeled log10(Teff/K) ranging from 3.30 to 4.90, and the y-axis is labeled MSE (log1p flux) ranging from 10^-6 to 10^-3. The plot shows median values and outliers across different effective temperatures. Panel D: Box plot for TMAP. The x-axis is labeled log10(Teff/K) ranging from 3.30 to 4.90, and the y-axis is labeled MSE (log1p flux) ranging from 10^-6 to 10^-3. The plot shows median values and outliers across different effective temperatures.

Figure 14.

Distribution of MSE across $\log g$ for the four libraries.

Figure 14. Long description

Panel A: The box plot for the PHOENIX-Husser library shows the distribution of mean squared error (MSE) in log1p flux across different values of log g. The x-axis represents log g values ranging from -1.00 to 8.50, and the y-axis represents MSE in log1p flux. The median MSE values are indicated by orange lines, and outliers are marked with gray circles. Panel B: The box plot for the C3K library shows the distribution of MSE in log1p flux across different values of log g. The x-axis represents log g values ranging from -1.00 to 8.50, and the y-axis represents MSE in log1p flux. The median MSE values are indicated by orange lines, and outliers are marked with gray circles. Panel C: The box plot for the OB–PoWR library shows the distribution of MSE in log1p flux across different values of log g. The x-axis represents log g values ranging from -1.00 to 8.50, and the y-axis represents MSE in log1p flux. The median MSE values are indicated by orange lines, and outliers are marked with gray circles. Panel D: The box plot for the TMAP library shows the distribution of MSE in log1p flux across different values of log g. The x-axis represents log g values ranging from -1.00 to 8.50, and the y-axis represents MSE in log1p flux. The median MSE values are indicated by orange lines, and outliers are marked with gray circles.

Figure 15.

Distribution of MSE across $\log Z$ for the two libraries with metallicities.

The largest errors and most prominent outliers occur at the lowest temperatures, roughly $3.3 \lesssim \log_{10}\kern-1.5pt(T_{\mathrm{eff}}/\mathrm{K}) \lesssim 3.4$ . This behaviour is expected because cool-star spectra are intrinsically more difficult to model than hotter, smoother spectra. In this regime, the emergent spectrum is shaped by dense molecular absorption bands and broad, highly structured opacity features, rather than by a relatively smooth continuum with isolated atomic lines. Small changes in $\log_{10}\kern-1.5pt(T_{\mathrm{eff}}/\mathrm{K})$ , $\log g$ , or $\log Z$ can therefore produce comparatively large changes in the detailed spectral morphology. As a result, interpolation in this region is more demanding, particularly for a neural network that is required to learn a single smooth manifold across a non-uniform multi-library grid.

The increased low-temperature error should therefore not be interpreted as a generic failure of SM-Net, but as a reflection of the difficulty of this part of the training domain. The lowest-temperature region is dominated by PHOENIX–Husser and C3K spectra, where the library coverage is non-uniform and the two libraries can differ substantially at neighbouring locations in the parameter grid. These differences arise from distinct atmosphere assumptions, opacity treatments, and line lists, and they are consistent with the cross-library residuals discussed in Section 5. Consequently, SM-Net is forced to learn a continuous representation through a region where the reference spectra themselves are more structured and less mutually consistent. Elevated outlier rates are also seen near library-overlap boundaries, especially around the PHOENIX–Husser/C3K transition. By contrast, regions dominated by a single internally consistent library, such as the hottest TMAP regime or the most metal-poor PHOENIX–Husser bins, show uniformly low errors and minimal scatter.

The per-library breakdown supports this interpretation. For $\log_{10}\kern-1.5pt(T_{\mathrm{eff}}/\mathrm{K})$ , the largest reconstruction errors at low temperature are concentrated in the PHOENIX–Husser and C3K libraries, which dominate the cool-star regime and contain strong molecular structure as well as library-dependent differences at fixed $(T_{\mathrm{eff}},\log g,\log Z)$ . At higher temperatures, where OB–PoWR and TMAP dominate, the MSE remains generally lower and prominent outliers are less common. For $\log g$ , the highest mean errors occur near $\log g=0$ for PHOENIX–Husser, where the MSE exceeds $\sim$ $10^{-4}$ , while the corresponding C3K errors remain close to $\sim$ $10^{-5}$ around $\log g=0$ . This reflects the denser local sampling of C3K in this region, which causes the learned manifold to follow the C3K spectral behaviour more closely and increases the residuals relative to the retained PHOENIX–Husser references. For $\log Z$ , the mean MSE remains broadly uniform across the two libraries with explicit metallicity coverage, with outliers dispersed rather than concentrated in any particular metallicity bin.

6.2. Synthetic broad-band colour and line-strength validation

The MSE values reported in Table 4 quantify reconstruction accuracy directly in the log1p-scaled flux representation used for training and evaluation. While this provides a measure of pixel-level spectral agreement, many astronomical applications rely on derived observables rather than individual flux samples. We therefore extend the validation in two complementary directions: synthetic broad-band colours, which probe the integrated spectral energy distribution over wide wavelength intervals, and line-strength indices, which probe the reconstruction of narrower spectral features.

Synthetic photometry is commonly used to evaluate stellar model fluxes, broad-band colours, and bolometric corrections (Casagrande & VandenBerg 2014). For this analysis, the spectra were transformed back to direct flux space using the inverse of the log1p scaling described in Section 2. Synthetic colours were then computed from band-averaged $F_\nu$ values using filter response curves from the Spanish Virtual Observatory (SVO) Filter Profile Service, an online repository of standardised photometric filter transmission curves (Rodrigo et al. Reference Rodrigo, Solano and Bayo2012, Reference Rodrigo2024; Rodrigo & Solano Reference Rodrigo and Solano2020). All colours are reported in the AB magnitude system, in which magnitudes are defined relative to a constant reference flux density per unit frequency.

We considered the colours $g-r$ , $r-i$ , $g-i$ , and $J-K_s$ . The first two probe adjacent optical bands, $g-i$ provides a broader optical baseline, and $J-K_s$ probes the near-infrared spectral shape. Together, the $(\,g-i)$ versus $(J-K_s)$ plane provides a combined optical and near-infrared colour–colour diagnostic. We use $K_s$ rather than the classical broad K band because it avoids the more problematic long-wavelength portion of the K response. The selected bands are representative of SDSS-like optical and 2MASS near-infrared photometry (Doi et al. Reference Doi2010; Skrutskie et al. Reference Skrutskie2006).

Figure 16 compares the retained test-set spectra and the corresponding SM-Net predictions in the $(\,g-i)$ versus $(J-K_s)$ plane. The contours are constructed from the full test set using a $42\times42$ two-dimensional histogram, lightly smoothed with a Gaussian kernel of width one histogram bin. They delineate the highest-density regions containing 50%, 80%, and 95% of the corresponding smoothed distribution, shown by the solid red, dashed blue, and dotted black contours, respectively. The same binning, smoothing, and colour–colour limits are applied to the reference and SM-Net distributions. The faint individual points are subsampled only for visual clarity and do not affect the contour calculation.

Figure 16.

Test-set colour–colour distributions in the $(\,g-i)_{\text{AB}}$ versus $(J-K_s)_{\text{AB}}$ plane for the retained reference spectra (left) and the corresponding SM-Net predictions (right). The Sun-like validation point is overlaid for reference. The solid red, dashed blue, and dotted black contours enclose, respectively, the highest-density regions containing 50%, 80%, and 95% of each smoothed distribution. The orange marker denotes the Sun-like validation point at $T_{\text{eff}}=5\,800\,$ K, $\log g=4.5$ , and $\log Z=0$ .

A Sun-like spectrum from the validation set is manually overlaid on both panels as an additional reference point. The corresponding SM-Net prediction lies close to the PHOENIX–Husser validation spectrum in colour–colour space. Figure 16 therefore provides a qualitative comparison of the global colour distributions, while the residual-based comparisons below quantify their reconstruction accuracy directly.

To quantify the colour-reconstruction precision directly, Figure 17 shows the residuals for $g-r$ , $r-i$ , $g-i$ , and $J-K_s$ as a function of $\log_{10}\kern-1.5pt(T_{\mathrm{eff}}/\mathrm{K})$ over the full test set. The running medians remain close to zero over most of the sampled temperature range, indicating little systematic colour bias, while the scatter increases towards the cooler end of the grid. The median absolute residuals are 0.022, 0.012, 0.028, and 0.015 mag for $g-r$ , $r-i$ , $g-i$ , and $J-K_s$ , respectively, with corresponding 90th-percentile absolute residuals of 0.070, 0.046, 0.077, and 0.063 mag. This full-test-set diagnostic complements Figure 16 by showing directly where the broad-band reconstruction errors occur in stellar-parameter space.

Figure 17.

Synthetic broad-band colour residuals across the full test set as a function of effective temperature. Residuals are defined as $\Delta(X-Y)=(X-Y)_{\text{reference}}-(X-Y)_{\text{SM-Net}}$ . The faint points show individual test spectra, while the solid line shows the running median and the shaded region denotes the 16–84 percentile interval. The median absolute residual and 90th-percentile absolute residual are reported in each panel.

Figure 17. Long description

Panel A: A scatter plot shows color residual in magnitudes for the color index g-r as a function of the logarithm of effective temperature in Kelvin. The faint points represent individual test spectra, while the solid blue line shows the running median and the shaded orange region denotes the 16-84 percentile interval. The median absolute residual is 0.022 mag, and the 90th percentile absolute residual is 0.070 mag. Panel B: A scatter plot shows color residual in magnitudes for the color index r-i as a function of the logarithm of effective temperature in Kelvin. The faint points represent individual test spectra, while the solid blue line shows the running median and the shaded orange region denotes the 16-84 percentile interval. The median absolute residual is 0.012 mag, and the 90th percentile absolute residual is 0.046 mag. Panel C: A scatter plot shows color residual in magnitudes for the color index g-i as a function of the logarithm of effective temperature in Kelvin. The faint points represent individual test spectra, while the solid blue line shows the running median and the shaded orange region denotes the 16-84 percentile interval. The median absolute residual is 0.028 mag, and the 90th percentile absolute residual is 0.077 mag. Panel D: A scatter plot shows color residual in magnitudes for the color index J-Ks as a function of the logarithm of effective temperature in Kelvin. The faint points represent individual test spectra, while the solid blue line shows the running median and the shaded orange region denotes the 16-84 percentile interval. The median absolute residual is 0.015 mag, and the 90th percentile absolute residual is 0.063 mag.

For the quantitative comparison with the intrinsic differences between stellar libraries, we next restrict the analysis to the 783 exact $(T_{\mathrm{eff}},\log g,\log Z)$ labels shared by PHOENIX–Husser and C3K. At each label, colours were calculated for the PHOENIX–Husser spectrum, the C3K spectrum, and the corresponding SM-Net prediction. This provides two directly comparable residual measurements over an identical stellar-parameter sample: PHOENIX–Husser versus C3K, which characterises the underlying cross-library variation, and PHOENIX–Husser versus SM-Net, which measures the additional reconstruction difference introduced by the model.

Using PHOENIX–Husser as the common reference, we define the PHOENIX–Husser/SM-Net colour difference as

(13) \begin{equation}\Delta_{\textrm {H-SM}}(X-Y) =(X-Y)_{\textrm {PHOENIX-Husser}}-(X-Y)_{\textrm {SM-Net}},\end{equation}

and the corresponding PHOENIX–Husser/C3K difference as

(14) \begin{equation}\Delta_{\textrm {H-C3K}}(X-Y) =(X-Y)_{\textrm {PHOENIX-Husser}}-(X-Y)_{\textrm {C3K}}.\end{equation}

A positive difference indicates that the PHOENIX–Husser colour is redder than the comparison colour, while a negative difference indicates that it is bluer. Both box plots are displayed over the common range $-0.5$ to $0.5$ mag for visual comparison. Values outside this displayed range remain included in the summary statistics.

Figure 18 shows that measurable library-to-library colour differences remain even when $T_{\mathrm{eff}}$ , $\log g$ , and $\log Z$ are identical. The PHOENIX–Husser/C3K distributions contain extended, predominantly negative tails, with the largest 90th-percentile absolute difference occurring for $g-i$ . These differences provide a reference scale for interpreting the corresponding SM-Net reconstruction residuals.

Figure 19 shows the PHOENIX–Husser/SM-Net differences over the same 783 labels. The signed medians remain close to zero, indicating little systematic colour bias, although the central SM-Net distributions are broader than the corresponding PHOENIX–Husser/C3K distributions.

To place the differences in the context of observational uncertainty, we used the band-level fractional flux error floors listed in Table 1 of Bellstedt et al. (Reference Bellstedt2020). These were converted to approximate magnitude uncertainties and the corresponding colour floor was calculated by adding the two band-level terms in quadrature:

(15) \begin{equation}\sigma_{X-Y} =\frac{2.5}{\ln 10}\sqrt{f_X^2+f_Y^2},\end{equation}

where $f_X$ and $f_Y$ are the fractional flux error floors in the two bands.

Figure 18.

Synthetic colour differences, defined as PHOENIX–Husser $-$ C3K, at the 783 exact matched stellar-parameter labels.

Figure 19.

Synthetic colour differences, defined as PHOENIX–Husser $-$ SM-Net, at the same 783 matched stellar-parameter labels.

Table 5 separates the typical differences, represented by the median absolute residual, from the upper-tail behaviour represented by the 90th percentile. The median absolute PHOENIX–Husser/C3K differences are smaller than the corresponding PHOENIX–Husser/SM-Net values for all four colours. For example, the median absolute $g-i$ differences are 0.0217 and 0.0307 mag, respectively. Conversely, the 90th-percentile SM-Net differences for $g-r$ , $r-i$ , and $g-i$ are smaller than the corresponding PHOENIX–Husser/C3K values, indicating less extended optical tails. For $J-K_s$ , SM-Net shows both a larger median absolute difference and a larger 90th-percentile difference than C3K, although its 90th-percentile value of 0.0787 mag remains below the adopted observational floor of 0.0997 mag. Overall, the matched-label comparison indicates that SM-Net does not simply reproduce either input library. Its central colour residuals are somewhat broader than the PHOENIX–Husser/C3K differences, while its extreme optical residuals are less extended.

Table 5.

Typical and upper-tail synthetic-colour differences at the 783 exact PHOENIX–Husser/C3K matched labels. Differences are defined relative to PHOENIX–Husser. The median absolute difference characterises the typical offset, while the 90th percentile characterises the upper tail. All values are in magnitudes.

Table 5. Long description

A table comparing synthetic-colour differences at matched labels. The table has four rows and six columns. Column headers are Median |Δ| Husser-SM-Net, Median |Δ| Husser-C3K, 90th percentile |Δ| Husser-SM-Net, 90th percentile |Δ| Husser-C3K, and Obs. floor. Row labels are g - r, r - i, g - i, and J - Ks. Row 1: Median |Δ| Husser-SM-Net, 0.0260; Median |Δ| Husser-C3K, 0.0147; 90th percentile |Δ| Husser-SM-Net, 0.0908; 90th percentile |Δ| Husser-C3K, 0.1219; Obs. floor, 0.0543. Row 2: Median |Δ| Husser-SM-Net, 0.0131; Median |Δ| Husser-C3K, 0.0084; 90th percentile |Δ| Husser-SM-Net, 0.0948; 90th percentile |Δ| Husser-C3K, 0.1357; Obs. floor, 0.0587. Row 3: Median |Δ| Husser-SM-Net, 0.0307; Median |Δ| Husser-C3K, 0.0217; 90th percentile |Δ| Husser-SM-Net, 0.0872; 90th percentile |Δ| Husser-C3K, 0.1961; Obs. floor, 0.0654. Row 4: Median |Δ| Husser-SM-Net, 0.0122; Median |Δ| Husser-C3K, 0.0091; 90th percentile |Δ| Husser-SM-Net, 0.0787; 90th percentile |Δ| Husser-C3K, 0.0484; Obs. floor, 0.0997.

Figure 20.

Comparison of line-strength indices from the reference test spectra and the corresponding SM-Net reconstructions for H $\beta$ , Fe5270, Fe5335, Ca4227, CN1, and TiO2. Atomic indices are given as pseudo-equivalent widths in Å, and CN1 and TiO2 in magnitudes. Points are coloured by $\log_{10}\kern-1.5pt(T_{\text{eff}}/\text{K})$ , with the dashed line indicating one-to-one agreement. Each panel reports r, $R^2$ , the median absolute residual, and the 90th percentile of the absolute residual.

Figure 20. Long description

Panel Hβ: A scatter plot compares the SM-Net index with the Reference index for Hβ. The x-axis is labeled ’Reference index [Å]’ and the y-axis is labeled ’SM-Net index [Å]’. The plot includes a dashed line indicating one-to-one agreement. Panel Fe5270: A scatter plot compares the SM-Net index with the Reference index for Fe5270. The x-axis is labeled ’Reference index [Å]’ and the y-axis is labeled ’SM-Net index [Å]’. The plot includes a dashed line indicating one-to-one agreement. Panel Fe5335: A scatter plot compares the SM-Net index with the Reference index for Fe5335. The x-axis is labeled ’Reference index [Å]’ and the y-axis is labeled ’SM-Net index [Å]’. The plot includes a dashed line indicating one-to-one agreement. Panel Ca4227: A scatter plot compares the SM-Net index with the Reference index for Ca4227. The x-axis is labeled ’Reference index [Å]’ and the y-axis is labeled ’SM-Net index [Å]’. The plot includes a dashed line indicating one-to-one agreement. Panel CN1: A scatter plot compares the SM-Net index with the Reference index for CN1. The x-axis is labeled ’Reference index [mag]’ and the y-axis is labeled ’SM-Net index [mag]’. The plot includes a dashed line indicating one-to-one agreement. Panel TiO2: A scatter plot compares the SM-Net index with the Reference index for TiO2. The x-axis is labeled ’Reference index [mag]’ and the y-axis is labeled ’SM-Net index [mag]’. The plot includes a dashed line indicating one-to-one agreement. Each panel reports r, R², the median absolute residual, and the 90th percentile of the absolute residual.

Broad-band colours primarily test the integrated spectral energy distribution and can remain accurate even when individual absorption features are imperfectly reconstructed. We therefore performed a complementary test using representative line-strength diagnostics over the full test set. The selected indices comprise the Balmer H $\beta$ feature, the Fe5270 and Fe5335 iron indices, Ca4227, and the molecular CN1 and TiO2 indices, thereby sampling hydrogen, metal, calcium, and molecular absorption features. The central feature bands used are 4847.875–4876.625 Å for H $\beta$ , 5245.650–5285.650 Å for Fe5270, 5312.125–5352.125 Å for Fe5335, 4222.250–4234.750 Å for Ca4227, 4142.125–4177.125 Å for CN1, and 6189.625–6272.125 Å for TiO2.

The atomic indices are measured as pseudo-equivalent widths in Å, whereas CN1 and TiO2 are measured in magnitudes. Identical feature and pseudo-continuum passbands are applied to each reference spectrum and its corresponding SM-Net reconstruction after transformation back to direct flux space. The pseudo-continuum is estimated from the blue and red sidebands surrounding each feature. Consequently, a negative atomic index indicates that the flux within the feature band lies, on average, above this fitted pseudo-continuum and does not by itself imply physical line emission. The line-strength residual is defined as $\Delta I=I_{\textrm {reference}}-I_{\textrm {SM-Net}}$ . The $R^2$ values shown in Figure 20 quantify agreement with exact one-to-one prediction rather than being defined simply as the square of the Pearson correlation coefficient. These diagnostics test the spectral features present in the underlying theoretical stellar libraries. They do not include additional line production associated with circumstellar or nebular gas and dust, which would require separate physical post-processing.

Figure 20 compares the indices measured from the reference spectra with those measured from the SM-Net reconstructions. The statistics in each panel are calculated using the full set of valid test spectra; the displayed points are subsampled only for visual clarity. Five of the six diagnostics show strong agreement with the one-to-one relation, with $R^2=0.931$ – $0.976$ for H $\beta$ , Fe5270, Fe5335, Ca4227, and TiO2. CN1 is less tightly reconstructed, with $R^2=0.696$ . For the atomic indices, the median absolute residual ranges from 0.173 Å for Fe5270 to 0.634 Å for H $\beta$ ; the corresponding median absolute residuals for CN1 and TiO2 are 0.0285 and 0.0185 mag. The temperature colouring also reveals parameter-dependent behaviour, with some of the larger deviations in the metal and molecular diagnostics associated with cooler spectra.

Additional per-library checks show that the behaviour is not uniform across the four input libraries. PHOENIX–Husser and C3K show strong line-index reconstruction, including the negative H $\beta$ branch already present in their reference spectra. TMAP also shows good H $\beta$ reconstruction, while several of its other indices span only a very small intrinsic range. The main weaker regime is OB–PoWR, particularly for H $\beta$ . This is consistent with the irregular spectral behaviour already identified during dataset quality control, where a subset of OB–PoWR spectra with strong discontinuities and abrupt continuum changes was removed. Even among the retained spectra, comparatively sharp local structure remains in some regions and is more difficult for SM-Net to reconstruct.

Appendix Figures A1 and A2 illustrate these effects directly. The first set of plots show the most extreme reference H $\beta$ case from each library together with the fitted pseudo-continuum, demonstrating how strong negative indices can arise in highly structured spectra. The second figure shows representative non-extreme cases in which both the H $\beta$ index and the local feature shape are reconstructed well, while also illustrating the greater difficulty of the OB–PoWR regime.

Taken together, the broad-band and line-strength analyses extend the evaluation beyond pixel-level MSE and test quantities that are more directly relevant to downstream astronomical applications. SM-Net reproduces the overall broad-band colour distributions with small systematic offsets and tracks several commonly used line-strength diagnostics closely, while the residual distributions, the weaker CN1 reconstruction, and the more challenging OB–PoWR line-index behaviour identify regimes in which additional uncertainty remains. These results therefore demonstrate both the utility and the current limitations of the learned spectral manifold for applications that depend on integrated colours or individual spectral features.

6.3. Reconstructed spectra

In this section, we present representative spectra reconstructed by SM-Net. The Sun-like example is drawn from the validation set, whereas the remaining examples are drawn from the test set. The validation example was not used for gradient-based training, while the test examples were not used during either training or checkpoint selection. All spectra are shown in the log1p-scaled flux representation used for training and evaluation. In each example, the upper panel shows the reference and SM-Net spectra, while the lower panel shows the residual, defined as reference $-$ SM-Net. The plotted axes are linear in this transformed flux space.

Figure 21 shows the spectrum in the combined dataset that is closest to the Sun, with parameters $T_{\mathrm{eff}} = 5\,800\,\mathrm{K}$ , $\log g = 4.5$ , and $\log Z = 0$ . This spectrum belongs to the validation set, and the prediction shown was generated using the best-validation checkpoint at epoch 389. The SM-Net prediction closely matches the corresponding PHOENIX–Husser template at the same parameter values. Most of the residual structure appears at shorter wavelengths; however, the peak deviations remain below $\sim$ $0.02$ in log1p-scaled flux, corresponding to less than 2% of the maximum normalised flux value of 1 in the transformed representation.

Figure 21.

Predicted spectrum for the closest Sun-like grid point, compared with the corresponding PHOENIX–Husser reference spectrum. The upper panel shows the spectra in log1p-scaled flux space, and the lower panel shows the residual, defined as reference $-$ SM-Net.

Figure 22.

High-fidelity SM-Net reconstruction for $T_{\text{eff}} = 6\,900\,\text{K}$ , $\log g = 4.0$ , and $\log Z = -2.0$ , shown against the corresponding PHOENIX–Husser reference spectrum.

Figure 22 shows the SM-Net reconstruction for a star with $T_{\mathrm{eff}}=6{\,}900\,\mathrm{K}$ , $\log g=4.0$ , and $\log Z=-2.0$ . The model accurately reproduces the fine spectral structure, achieving a mean-squared error of $1.44\times10^{-6}$ relative to the PHOENIX–Husser reference spectrum. This example illustrates the high reconstruction accuracy achieved in this region of parameter space.

Figure 23 presents another high-quality reconstruction, this time for a hotter star with $T_{\mathrm{eff}}=13{\,}000\,\mathrm{K}$ , $\log g=4.5$ , and $\log Z=-1.5$ . SM-Net reproduces the detailed spectral structure with a mean-squared error of $9.67\times10^{-7}$ relative to the corresponding C3K reference spectrum. Together, Figures 22 and 23 provide high-quality examples drawn from two different constituent libraries.

Figure 24 shows an example of a poor-quality reconstruction for $T_{\mathrm{eff}} = 2\,200\,\mathrm{K}$ , $\log g = 3.0$ , and $\log Z = -2.1$ , compared against the corresponding C3K reference spectrum. The model exhibits substantial residuals across the full wavelength range, with the largest deviations occurring at shorter wavelengths. This case highlights the greater difficulty of accurately reconstructing spectra in the very low–temperature regime.

Figure 23.

High-quality SM-Net reconstruction for $T_{\text{eff}} = 13{\,}000\,\text{K}$ , $\log g = 4.5$ , and $\log Z = -1.5$ , compared with the corresponding C3K reference spectrum.

Figure 24.

Example of a low-quality reconstruction at $T_{\text{eff}} = 2\,200\,\text{K}$ , $\log g = 3.0$ , and $\log Z = -2.1$ , shown against the corresponding C3K reference spectrum.

Figure 25 presents another example of a low-quality reconstruction for $T_{\mathrm{eff}}=22{\,}000\,\mathrm{K}$ , $\log g=2.4$ , and $\log Z=-1.0$ , compared with the corresponding OB–PoWR reference spectrum. SM-Net struggles to reproduce both the spectral features and the overall wavelength-dependent slope, with the residual reaching approximately 0.04 in the normalised log1p-scaled flux representation at the longest wavelengths. This example illustrates the difficulty of modelling regions in which sparse library coverage makes interpolation particularly challenging.

Figure 25.

Example of a low-quality reconstruction for $T_{\text{eff}} = 22\,000\,\text{K}$ , $\log g = 2.4$ , and $\log Z = -1.0$ , compared against the corresponding OB–PoWR reference spectrum.

Figure 26.

Example of SM-Net reconstructing the missing high-wavelength flux for $T_{\text{eff}} = 60{\,}000\,\text{K}$ and $\log g = 6.5$ , compared with the corresponding TMAP reference spectrum.

Figure 26, corresponding to $T_{\mathrm{eff}}=60{\,}000\,\mathrm{K}$ , $\log g=6.5$ , and an assigned $\log Z=1.0$ , illustrates SM-Net’s reconstruction of a masked high-wavelength region in a TMAP spectrum. The model continues the spectral trend through the region in which the reference flux is unavailable while remaining consistent with the surrounding continuum. Because TMAP does not explicitly parametrise metallicity, the assigned $\log Z$ value serves only to embed the spectrum in the common three-parameter grid. This example demonstrates how SM-Net uses correlations learned from neighbouring grid points to infer a smooth continuation through a masked wavelength interval.

Across these examples, reconstruction quality is strongly dependent on the local spectral regime. The best PHOENIX–Husser, C3K, and TMAP cases achieve MSE values of approximately $10^{-6}$ , demonstrating that SM-Net can reproduce both continuum shape and detailed spectral features with high accuracy in well-represented regions of the grid. The contrasting low-quality examples show that errors increase in sparsely sampled or physically complex regimes, particularly at very low temperatures and near difficult library boundaries. The TMAP example additionally demonstrates that the learned model can infer a smooth continuation through wavelength regions masked in the reference spectrum. These results are consistent with the parameter-dependent error distributions discussed in Section 6.1 rather than indicating uniform performance throughout the combined grid.

SM-Net can also be evaluated at parameter combinations outside the convex hull of the training data, producing smooth extrapolative predictions based on trends learned from neighbouring regions of parameter space. However, these predictions are not directly validated by reference spectra and should be treated as exploratory rather than physically established. Consequently, extrapolated spectra should be interpreted with caution and regarded as hypotheses generated by the learned manifold rather than as physically validated models. Systematic assessment of the physical reliability of extrapolated predictions is left to future work.