A Physics Informed Neural Network For Deriving MHD State Vectors From Global Active Regions Observations

TL;DR

The study tackles the challenge of deriving physically consistent MHD tachocline initial conditions from surface AR toroid patterns to enable weeks-ahead predictions. It introduces PINNBARDS, a Physics-Informed Neural Network that couples observed toroid geometry with nonlinear MHD shallow-water equations to infer state-vectors (u,v,a,b,h) at t=0. The results show convergence to predominantly antisymmetric toroidal bands that closely match observed patterns, with best agreement for intermediate toroidal field strengths in the range $20$-$30$ kG and latitudinal bandwidth around $10^b$, under subadiabatic tachocline conditions ($G\,\approx\,0.5$). The method demonstrates robustness across solar-cycle phases and extends to other magnetogram datasets, offering a first pathway to reconstruct hidden tachocline magnetic structures and to seed forward models for potential weeks-long AR emergence prediction.

Abstract

Solar active regions (ARs) do not appear randomly but cluster along longitudinally warped toroidal bands ('toroids') that encode information about magnetic structures in the tachocline, where global-scale organization likely originates. Global MagnetoHydroDynamic Shallow-Water Tachocline (MHD-SWT) models have shown potential to simulate such toroids, matching observations qualitatively. For week-scale early prediction of flare-producing AR emergence, forward-integration of these toroids is necessary. This requires model initialization with a dynamically self-consistent MHD state-vector that includes magnetic, flow fields, and shell-thickness variations. However, synoptic magnetograms provide only geometric shape of toroids, not the state-vector needed to initialize MHD-SWT models. To address this challenging task, we develop PINNBARDS, a novel Physics-Informed Neural Network (PINN)-Based AR Distribution Simulator, that uses observational toroids and MHD-SWT equations to derive initial state-vector. Using Feb-14-2024 SDO/HMI synoptic map, we show that PINN converges to physically consistent, predominantly antisymmetric toroids, matching observed ones. Although surface data provides north and south toroids' central latitudes, and their latitudinal widths, they cannot determine tachocline field strengths, connected to AR emergence. We explore here solutions across a broad parameter range, finding hydrodynamically-dominated structures for weak fields (~2 kG) and overly rigid behavior for strong fields (~100 kG). We obtain best agreement with observations for 20-30 kG toroidal fields, and ~10 degree bandwidth, consistent with low-order longitudinal mode excitation. To our knowledge, PINNBARDS serves as the first method for reconstructing state-vectors for hidden tachocline magnetic structures from surface patterns; potentially leading to weeks ahead prediction of flare-producing AR-emergence.

Figures (12)

• Figure 1: Two warped toroid patterns displayed in North (blue) and South (red) hemispheres. Solid curves in blue and red respectively denote the central latitudes of the AR distributions in North and South; blue (red) dashed curves on both sides of solid blue (red) curve imply the width of the toroidal band in which ARs are strung. Absolute longitude (Carrington longitude) is shown on the top x-axis, while the bottom x-axis will be used in PINN model, and later on PINN-solutions will be compared with observations according to bottom x-axis.
• Figure 2: Workflow of the Physics Informed Neural Network (PINN) for an MHD shallow-water model. Latitude–longitude coordinates ($\phi$-$\lambda$) are converted to Cartesian coordinates ($x,y,z$) and fed into a neural network that predicts the physical state variables ($a, b, u, v, h$), namely longitude and latitude components of magnetic fields ($a,b$) and velocity fields ($u,v$) and $h$ is related to thickness of the thin-shell as $1+h$. The model is trained using observational data and constrained by the full set of nonlinear magnetohydrodynamic equations, including mass conservation and divergence-free magnetic field conditions.
• Figure 3: Sampling of latitudes for PINN training. The transformation on the left converts $\phi_u$ to $\phi$ (red points), lowering the latitudes away from the poles ('y=x' is added for reference). This converts, as shown on the right panel, the uniform distribution (histogram density in light blue) of $\phi_u$ in $[-\pi/2, \pi/2]$ to a cosine-modulated distribution (histogram density in light orange) that follows the distribution of latitudes when points are uniformly sampled from the surface of a sphere.
• Figure 4: Convergence of PINN and identification of optimal architecture. Plot in panel (a) shows the evolution of PINN loss with epoch for the architecture with 10 layers and 100 nodes/layer. The loss is a combination of equation loss and data loss. The blue curve shows the loss for all the epochs, whereas the orange curve shows the monotonic version with epochs ($e$) defined by the set $S = \{\, e \mid Loss(e) \le \min_{k = 0}^{e - 1} Loss(k) \,\}$. We highlight the PINN convergence for a set of four epochs, marked in filled red circles, by visualizing the state vector outcomes in Figure \ref{['fig:pinn_cnvrg']}. Panel (b) shows how the minimum loss (Loss$_{min}$) changes when the number of layers ($N_l$) and the number of nodes/layer ($N_d$) are changed from 10 and 100, respectively. It can be seen that for a much smaller $N_d$, such as 20, the loss is larger for all $N_l$, causing less accurate results. However for $N_d\geq80$ and $N_l\geq10$ the Loss$_{min}$ goes below $10^{-4}$ in an asymptotic manner, and this difference is likely due to statistical fluctuation from model initialization. We thus stick to an $N_d=100$ and $N_l=10$ as an optimal PINN architecture in terms of speed and accuracy. We perform all further experiments with the same architecture.
• Figure 5: PINN-simulated MHD shallow-water solutions to replicate the observed warped toroidal magnetic bands of February 14, 2024. Left panels: magnetic field vectors (black arrows) overlaid on height-deformation contours ($h-<h_\phi>$) in rainbow color-map, where red (blue) indicates bulging (depression); right panels: magnitude of toroidal bands ($\sqrt{a^2+b^2}$). These two columns from top to the bottom display how the PINN progressively converges from broad, unstructured fields (high loss) to well-defined, antisymmetric toroidal bands that closely reproduce the observed longitudinal warping, demonstrating convergence of solution for global magnetic topology consistent with solar observations.