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Physics-Informed Neural Networks for Fast Multilayer Spectral Inversion of H{\alpha} 6562.8 A and Ca II 8542.1 A Spectra
Ziyang Zhang, Qin Li, Vasyl B. Yurchyshyn, Kangwoo Yi, Haimin Wang, Wenda Cao, Bo Shen
TL;DR
Conventional MLSI provides interpretable multilayer inversions of chromospheric spectra, but pixel-by-pixel nonlinear fitting is costly. This work embeds the analytic MLSI forward model in a two-stage neural network, which reproduces main parameter structures and spectra comparable to direct MLSI synthesis in quiet-Sun and active-region FISS observations.
Problem
Conventional MLSI requires constrained nonlinear least-squares fitting at each pixel, making application to many FISS rasters or long time sequences costly.
Source: §1 · p. 2Method
A neural network predicts MLSI parameters from observed profiles and uses the analytic forward model for spectral reconstruction, then fine-tunes selected parameters using a limited conventional MLSI reference raster.
Source: §1 · p. 3Results
Applied to quiet-Sun and active-region FISS observations, MLSI-PINN reproduces conventional MLSI’s main parameter structures and reconstructed spectra with residuals comparable to direct MLSI forward synthesis.
Source: §5 · p. 20Takeaways & Limitations
The framework is an acceleration strategy for MLSI-style analysis of large FISS datasets and time-dependent observations, rather than a replacement for detailed radiative-transfer inversions.
Source: §1 · p. 3Stage-2 parameter supervision inherits assumptions and possible biases of the conventional reference inversion, and simplified MLSI cannot replace full NLTE inversions when detailed stratification, magnetic-field information, or more realistic radiative transfer is required.
Source: §5 · p. 21
Abstract
from arXiv · showhide
Strong chromospheric absorption lines such as H$\alpha$ 6562.8 A and Ca II 8542.1 A provide vital diagnostics of plasma dynamics and thermal structure in the solar chromosphere. Multilayer spectral inversion (MLSI) offers a physically interpretable framework for modeling these lines using a finite number of radiative-transfer layers, but conventional MLSI relies on pixel-by-pixel nonlinear least-squares fitting, making it computationally expensive for large imaging spectroscopic data sets. Here, we introduce a physics-informed neural-network (PINN) framework to accelerate MLSI while preserving its analytic radiative-transfer formulation. The network predicts MLSI parameters directly from observed line profiles and passes them through a differentiable MLSI forward model to synthesize spectra. Training follows a two-stage approach: an initial stage optimized solely via spectral reconstruction loss, followed by fine-tuning that combines spectral consistency with parameter-space supervision from conventional MLSI results on a single reference image. This strategy eliminates the need for large precomputed training sets while maintaining physical interpretability. Applied to Fast Imaging Solar Spectrograph (FISS) observations from the Goode Solar Telescope (GST) targeting both quiet-Sun and active-region regions, MLSI-PINN parameter maps reproduce the primary spatial structures of direct inversions, achieving an arithmetic mean pixel-wise Pearson correlation coefficient of 0.933 across all evaluated parameters. The reconstructed spectra closely match both observed profiles and conventional MLSI fits. Post-training, MLSI-PINN processes a raster in approximately 5-15 seconds compared to 3-5 minutes for conventional MLSI, delivering an inference speedup of about 12-60 times without substantial loss in reconstruction quality, enabling efficient MLSI analysis on large chromospheric data sets.
1. INTRODUCTION
Hα and Ca II spectra encode thermal and dynamic information, but extracting atmospheric parameters requires inversion. MLSI offers a physically interpretable multilayer model, yet repeated pixel-by-pixel fitting remains costly for large datasets; this work embeds its analytic forward model in a two-stage neural-network framework.
1. INTRODUCTION
- Repeated forward-model evaluations and pixel-by-pixel fitting can dominate analysis costs for high-cadence rasters and long observing sequences.More physically detailed inversion models generally require more expensive radiative-transfer calculations and nonlinear optimization.Source: §1 · p. 2
- Conventional MLSI models photospheric and chromospheric contributions with finite radiative-transfer layers, but still requires constrained nonlinear least-squares fitting at each pixel.It is lighter computationally than stratified NLTE inversion, while remaining costly across many FISS rasters or long sequences.Source: §1 · p. 2
- Prior supervised MLSI networks infer parameters quickly but require sufficiently large, representative conventional inversion labels.Their performance depends on training-label coverage and consistency, and may inherit reference-inversion ambiguities or biases.Source: §1 · p. 2
- MLSI-PINN maps observed FISS Hα and Ca II 8542 profiles to MLSI parameters and constrains them through the analytic forward model.A second training stage adds parameter-space supervision from selected quantities in a limited conventional MLSI reference raster.Source: §1 · p. 3
- The proposed approach is framed as an acceleration strategy for large FISS datasets and time-dependent observations, not a replacement for detailed radiative-transfer inversions.The framework focuses on intensity-only spectroscopic inversion.Source: §1 · p. 3
2. METHOD
The method combines an analytic multilayer radiative-transfer model with a neural network that maps observed profiles to MLSI parameters. Training first uses spectral reconstruction alone, then adds selected conventional MLSI parameter references to refine the predictions.
2.2. Physics-Informed Neural Network Framework
- MLSI-PINN reproduces conventional MLSI’s main parameter structures and yields reconstructed spectra with residuals comparable to direct MLSI forward synthesis.The forward model remains inside training, constraining predicted parameters to reproduce observed spectra.Source: §2.1 · p. 3
2.1. MLSI Forward Model
- MLSI represents a continuous atmosphere with a photospheric layer and two chromospheric layers, yielding an analytic spectrum from a finite set of interpretable parameters.The model uses a Voigt-like photospheric background and applies two chromospheric layers to it.Source: §2.1 · p. 3
- The network maps a normalized observed profile to MLSI parameters, constrains them to physically allowed ranges, and passes them through the analytic operator to synthesize a spectrum.The photospheric background is propagated through the lower and then upper chromospheric layer to form the emergent profile.Source: §2.1 · p. 4, §2.2 · p. 5
2.2. Physics-Informed Neural Network Framework
- Stage 1 trains the profile-to-parameter mapping using spectral reconstruction loss alone, without requiring a large set of precomputed MLSI reference parameters.Different parameter combinations can produce similar profiles, so this spectral constraint may not uniquely determine every parameter.Source: §2.2 · p. 5
- Stage 2 initializes from stage 1 and adds parameter-space supervision from selected conventional MLSI values in a single reference raster.The combined objective retains spectral consistency while refining parameters selected for systematic deviations; α and β weight the two constraints.Source: Figure 1 · p. 4, §2.2 · p. 5
- After training, inference uses one neural-network forward pass; the analytic MLSI calculation is also performed when reconstructed spectra are required.The Hα active-region example uses a fully connected network with hidden layers of 256, 128, and 64 neurons.Source: §2.2 · pp. 5–6
3. DATA
The study uses two FISS observing sequences covering quiet-Sun and active-region conditions, then calibrates and preprocesses their spectra for MLSI-PINN training and evaluation.
3.1. FISS Observations
- FISS observations use approximately 0.16″ spatial sampling and raster cadences of approximately 20–35 seconds.Spectral coverage is approximately 9.7 Å for Hα and 12.9 Å for Ca II 8542, with sampling of about 0.019 Å and 0.026 Å, respectively.Source: §3.1 · p. 6
- The quiet-Sun sequence contains 105 frames, while the active-region sequence contains 83 frames; both include simultaneous Hα and Ca II 8542 spectra.The sequences were observed on 2021 August 7 and 2023 August 14, respectively.Source: §3.1 · p. 6
- The quiet-Sun and active-region processed cubes measure 502 × 246 × 200 and 512 × 256 × 150, respectively, in wavelength, slit-position, and scan-position axes.Table 1 summarizes the observing sequences.Source: §3.1 · p. 6, Table 1 · p. 6
3.2. Data Reduction and Preprocessing
- Preprocessing removes detector signatures, corrects spectrogram geometry, and applies wavelength calibration to produce calibrated spectral cubes.The correction rectifies the wavelength and spatial axes before assigning physical wavelengths to spectral channels.Source: §3.2 · pp. 6–7
- Hα and Ca II 8542 are processed separately on fixed wavelength grids, using only intensity profiles as network inputs.The fixed grids keep the input dimension fixed and align the MLSI forward operator with observed wavelength samples.Source: §3.2 · p. 7
- One reference image per sequence and line trains the first-stage model and supplies conventional MLSI parameter targets for second-stage supervision.The remaining frames are reserved for inference and evaluation, not network optimization.Source: §3.2 · p. 7
- Each profile is normalized by its mean intensity across selected line-continuum samples, bringing continuum intensities typically close to unity.The same normalization is used for reconstruction loss and inference and improves neural-network training conditioning.Source: §3.2 · p. 7
4. RESULTS
The results evaluate MLSI-PINN on quiet-Sun and active-region FISS observations, examining its parameter structures, spectral reconstruction, and post-training raster processing, as well as its ability to reconstruct broad absorption profiles from IAG atlas spectra. The atlas comparison tests spectral representation rather than the accuracy of inferred MLSI parameters.
4.1. FISS Observations and Spectral Morphology
- Active-region FISS profiles show a sunspot, stronger intensity gradients, and more structured line cores than quiet-Sun profiles, spanning broader ranges of line depths, Doppler shifts, and asymmetries.The observations sample seven wavelength offsets from −4 to +4 ˚A; the wavelength-dependent profiles motivate quantitative inversion using the full spectrum rather than a single image.Source: §4.1 · p. 7, Figure 2 · p. 8
4.2. Parameter-Space Agreement with Conventional MLSI
- Representative-raster parameter CC values span 0.774–1.000 for quiet Sun and 0.812–1.000 for active regions, with strongest agreement for source functions.Chromospheric velocities and selected Doppler widths show greater scatter because of line-profile sensitivity and multilayer degeneracy; narrow width ranges can also make R2 sensitive to small offsets.Source: Figure 5 · p. 11, Figure 6 · p. 12, §4.2 · pp. 12–13
4.3. Spectral Reconstruction and Temporal Evolution
- During observed intervals, MLSI-PINN follows quiet-Sun and active-region line-profile changes and reproduces temporal trends in upper-layer velocities and derived temperature and nonthermal velocity.The active-region sequence contains observing gaps that remain un-interpolated; temperature and nonthermal velocity are derived from upper-layer widths under shared-plasma and thermal-plus-nonthermal broadening assumptions.Source: §4.3 · p. 16
- Across the four sequence cases, conventional MLSI mean absolute residuals range from approximately 5.9 × 10−3 to 7.7 × 10−3, versus approximately 1.1 × 10−2 to 1.5 × 10−2 for MLSI-PINN.The residuals remain on the same order of magnitude; small wavelength shifts can increase fixed-sample residuals, especially on steep line-core flanks, without substantially changing overall profile morphology.Source: §4.3 · pp. 16, 19, Figure 13 · p. 19
4.4. Computational Performance
- After one-time training, MLSI-PINN processes each additional raster in several seconds, versus several minutes for CPU-based pixel-by-pixel conventional MLSI fitting.This is a practical wall-clock comparison, not a hardware-matched benchmark; training cost is amortized over sequences of tens to hundreds of rasters.Source: §4.4 · p. 19, Table 3 · p. 19
4.5. Spectral Reconstruction Using the Institut f¨ur Astrophysik G¨ottingen (IAG) Solar Flux Atlas
- Without stage-2 parameter supervision, stage-1 MLSI-PINN reconstructs the broad Hα and Ca II 8542 absorption profiles from resampled IAG solar flux atlas spectra.Mean residuals ⟨I0 − Iobs⟩ over clean samples within ±4 ˚A are 4.99 × 10−3 for Hα and −6.53 × 10−3 for Ca II 8542; localized peaks reflect narrow features absent from the compact MLSI model.Source: Figure 14 · p. 20, §4.5 · p. 20
- The framework adapted to high-resolution IAG solar flux spectra using its analytic multilayer construction, but the comparison evaluates spectral representability rather than inferred-parameter accuracy.The atlas provides disk-integrated solar flux, not reference atmospheric parameters.Source: §4.5 · p. 20
5. SUMMARY AND DISCUSSION
The two-stage MLSI-PINN retains the analytic MLSI forward model while accelerating inversion of Hα and Ca II 8542 spectra. On quiet-Sun and active-region FISS observations, it reproduces the main parameter structures and spectra of conventional MLSI, with validation focused on same-day or same-sequence use.
5. SUMMARY AND DISCUSSION
- On quiet-Sun and active-region FISS observations, MLSI-PINN reproduces conventional MLSI’s main parameter structures and yields reconstructed spectra with comparable residuals.The predicted parameters are constrained to reproduce observed spectra through the analytic multilayer radiative-transfer formulation.Source: §5 · p. 20
- Stage 1 uses spectral reconstruction loss alone, while stage 2 adds limited conventional-MLSI parameter supervision for selected parameters alongside that loss.The second stage guides parameters that are less strongly constrained by line profiles without requiring a large precomputed training set.Source: §5 · p. 21
- Source functions are more robust than chromospheric velocities and some Doppler widths, which are harder to constrain because of subtle profile asymmetries and parameter degeneracy.The photospheric velocity vp is inserted from a line-center proxy rather than independently learned by the network.Source: §5 · p. 21
- Validation focuses on same-day or same-sequence use, and the method is not presented as transferable across instruments, observing conditions, or substantially different calibration states without adjustment.The authors describe retraining or fine-tuning for a new observing day as practical.Source: §5 · p. 21
- Stage-2 parameter supervision inherits conventional MLSI assumptions and possible biases, and simplified MLSI cannot replace full NLTE inversions when detailed stratification, magnetic fields, or more realistic radiative transfer are required.The authors identify broader validation, uncertainty quantification for degenerate parameters, and transferability testing as future work.Source: §5 · p. 21