2.1. Model training

For training the neural network, synthetic data for temperature, T, density, $\rho$ , and both velocity and magnetic field in the line-of-sight (LOS), $v_{\text{LOS}}$ and $B_{\text{LOS}}$ , were generated using the MURaM simulation code Vögler et al. (Reference Vögler, Shelyag, Schüssler, Cattaneo, Emonet and Linde2005) in a domain that includes both the convective zone and the photosphere, as seen in Fig. 1. In this domain, the atmospheric parameters were generated along geometric height. However, since actual inversion methodologies generate the corresponding atmospheric parameters using optical depth as a measurement of height, it is necessary to perform a mapping from geometrical height to optical depth. This process is applied by using the Rosseland opacities Rose (Reference Rose2001), obtaining 21 optical depth values ranging from $\log \tau = -2.0$ to $\log \tau = 0$ with a sampling of $\log \tau = 0.1$ . Apart from the atmospheric parameters, we also have the corresponding spectropolarimetric profiles for each Stokes parameter I, Q, U, and V, which were generated by applying the NLTE numerical code NICOLE (Socas-Navarro et al. Reference Socas-Navarro, de la Cruz Rodríguez, Asensio Ramos, Trujillo Bueno and Ruiz Cobo2015) for 300 wavelength points from $6300.5\,\unicode{x212B}$ to $6303.5\,\unicode{x212B}$ with a spectral sampling of $\Delta \lambda = 0.01\,\unicode{x212B}$ , including the FeI absorption lines in $6301.5\,\unicode{x212B}$ and $6302.5\,\unicode{x212B}$ . This data was adapted to the Hinode SOT/SP instrument specifications (Tsuneta et al. Reference Tsuneta, Ichimoto, Katsukawa, Nagata, Otsubo, Shimizu, Suematsu, Nakagiri, Noguchi, Tarbell, Title, Shine, Rosenberg, Hoffmann, Jurcevich, Kushner, Levay, Lites, Elmore, Matsushita, Kawaguchi, Saito, Mikami, Hill and Owens2008) by applying the spectral and spatial sampling, the spectral and spatial PSF, and the noise of the instrument, corresponding to gaussian random noise with a $\sigma = 5 \times 10^{-3}$ of the continuum intensity $I_c$ . Additionally, a weighting was applied to the stokes parameters, realizing that there are differences of orders of magnitude (Lagg et al. Reference Lagg, Solanki, Doerr, Martnez González, Riethmüller, Collados Vera, Schlichenmaier, Orozco Suárez, Franz, Feller, Kuckein, Schmidt, Asensio Ramos, Pastor Yabar, von der Lühe, Denker, Balthasar, Volkmer, Staude, Hofmann, Strassmeier, Kneer, Waldmann, Borrero, Sobotka, Verma, Louis, Rezaei, Soltau, Berkefeld, Sigwarth, Schmidt, Kiess and Nicklas2016; Castellanos Durán et al. Reference Castellanos Durán, Milanovic, Korpi-Lagg, Löptien, van Noort and Solanki2024) between them, which could lead the neural network to prioritize those with higher magnitudes if not treated properly. Also, We applied a filtering technique using the Otsu threshold (Otsu Reference Otsu1979) to separate both regions with high and low intensity, as well as the regions with high and low polarization This approach allows us to have an equal amount of data from both regions, despite their physically different behaviors (Vögler et al. Reference Vögler, Shelyag, Schüssler, Cattaneo, Emonet and Linde2005).

Figure 1.

Domain of the MURaM simulated parameters. The domain includes the corrugated surface where the optical depth, $\tau$ , meets the condition $\log \tau = 1$ , meaning that the domain includes both convective zone and photosphere from the solar atmosphere. The geometrical dimensions of the box are $12\,\text{Mm} \times 12\,\text{Mm}$ with a height of $2.56\,\text{Mm}$ .

The data extracted correspond to various simulation steps of the MURaM simulation for the quiet sun as observed in the center of the disk. After applying all the preprocessing, we obtained a total of $140\,382$ pixels for training. For testing the model on synthetic data, a step of the simulation was left outside the training set.

2.2. Observational

To test the ability of generalization of the neural network model, it is intended to apply it to actual observations. In this case, an observation corresponding to the MODEST catalog (Castellanos Durán et al. Reference Castellanos Durán, Milanovic, Korpi-Lagg, Löptien, van Noort and Solanki2024), with ID 0560, was chosen for testing. This observation corresponds to Active region number 11967 from the NOAA Active Region catalog, observed on February 3, 2014, with a field of view of $254'' \times 124''$ , longitude $x_c = 16''$ , latitude $y_c''=126$ , and heliocentric angle $\mu=0.99$ , indicating that we are close to the center of the disk. Additionally, based on this observation, we selected a region that did not include the umbra or penumbra. An image on the continuum of this AR, along with the selected region, is shown in Fig. 2. The atmospheric parameters of this observation in the MODEST catalog correspond to SPINOR 2D spatially coupled inversions (Siu-Tapia et al. Reference Siu-Tapia, Lagg, Solanki, van Noort and Jurčák2017).

Figure 2.

Observation with ID 0560 from the MODEST catalog along with the selected region used for testing the neural network model.

3.1. Architecture

Since the data we are dealing with correspond to spectropolarimetric profiles, it is necessary to use a model that can extract as many features as possible from a 1D signal. For this purpose, we used a mixture of convolutional neural networks (CNNs) with deeply connected layers in an architecture named Multi-Scale Convolutional Neural Network (MSCNN; see Fig. 3), which was first implemented by Jiang et al. (Reference Jiang, He, Yan and Xie2019) with the aim of diagnosing turbine gearbox health through the analysis of the turbine time series. This architecture implements a coarse-grain layer that separates the profiles in different scalings through the application of an average kernel. Each scale passes through a convolutional block that extracts a 1D feature map. All the feature maps are concatenated and then provided to a fully connected block that retrieves the corresponding atmospheric parameters for each analyzed pixel in its outer layer.

Figure 3.

Scheme of the MSCNN architecture. The model receives the weighted stokes parameters, coarse-grain and generate feature maps for each scale for finally passing the concatenated maps to a fully connected neural network block that leads to the output of the corresponding atmospheric parameters.

Figure 3 Long description

A diagram of the MSCNN architecture. The model starts with the input of weighted Stokes parameters, represented as four different graphs. These parameters undergo coarse-grain processing, depicted in the green section. The processed data then moves through multiple convolutional blocks, shown in the orange section, generating feature maps at different scales. These feature maps are concatenated and flattened before being passed into a series of fully-connected layers, illustrated in the yellow section. The output of the fully-connected layers provides the corresponding atmospheric parameters, labeled as T, log ρ, v_LOS, and B_LOS.

For training the neural network, an Adam optimizer was used, along with a learning rate of $10^{-3}$ and 40 epochs of training.

3.2. Physical constraint

Based on the idea of Physics-Informed Neural Networks (PINNs; Raissi et al. (Reference Raissi, Perdikaris and Karniadakis2019), we modified the loss function to include a term based on the Weak Field Approximation (WFA) for the $B_{\text{LOS}}$ . The total loss function uses a weight w to leverage the influence of the terms, where the WFA term is multiplied by w while the mean absolute error (MSE) term is multiplied by $1-w$ ). The WFA constraint term compares the magnitude approximation for the magnetic field in the line-of-sight with specific optical depth layers that present the most similarity to the results of the $B_{\text{LOS}}$ obtained by the WFA. The approximation is only applied under a threshold of WFA $B_{\text{LOS}}$ under $1000\,$ G and also after the MSE loss term reaches values below $4 \times 10^{-4}$ . The values of w used for the experiment were $w = 0, 0.001, 0.01, 0.1, 0.5$ .