Neural Differential Equations for the Solar Dynamo

Abstract

Physical models aimed to reproduce basic features of the solar sunspot cycle are typically based on the solar dynamo mechanism. Usually qualitative arguments are used to define parameters of the model, among which a challenging component is the nonlinear form of quenching of the alpha-effect governing regeneration of the magnetic field. We propose a novel approach, in which the functional form of the alpha-quenching is represented by a neural network model embedded into neural differential dynamo equations trained on observational data. For demonstration, we consider a low-mode dynamo model and find a wide set of alpha-quenching functions and corresponding dynamo numbers that provide an accurate fit to the average profile of the solar cycle data given by sunspot numbers. Within this set, we observe a strong relationship between the dynamo number and the shape of the alpha-quenching function indicating that additional magnetic field data or constraints are essential to unambiguously infer parameters of the dynamo model. In our opinion, the neural differential approach opens a new prospect for data-driven investigation of the closure problem in dynamo theory.

Figures (5)

• Figure 1: Scheme of the neural differential framework. Green boxes contain parameters to be optimized. Current values of these parameters are passed to the dynamo model and the ODE solver yields the numerical solutions $A$, $B$. Numerical solutions are compared with observational data and the loss function is computed. Based on the dynamo ODE, solutions $A$, $B$, and the loss function, the adjoint ODE is constructed (see Appendix A for an example). Integration of the adjoint ODE using the ODE solver yields the gradients of the loss function with respect to the model parameters. Optimizer uses the gradients to update the model parameters.
• Figure 2: One realization of the reconstruction of the dynamo parameters when only $ReB^2$ is fitted. Upper panel shows a comparison of the initial state of the model (green line), fitted model (orange line) and target (blue line) values of $ReB^2$. Bottom panel shows the initial function $\alpha$ (green line), reconstructed (orange line) and ground-truth (blue line) function $\alpha$. Dashed gray line shows the maximum value of $|ReB|$ after stabilization of the dynamo model.
• Figure 3: Upper panel: variability (two-sigma interval) of the fitted models (orange area) and the ground-truth values of $ReB^2$. Middle panel: variability (two-sigma intervals) of reconstructed functions $\alpha$ when (i) only $ReB^2$ is fitted (gray area), (ii) both $ReB^2$ and $ReA$ are fitted (green area), and (iii) full set $ReA$, $ImA$, $ReB$, $ImB$ is fitted. Black line shows the ground-truth function $\alpha(B)=1/(1+ReB^2)$. Bottom panel: individual reconstructed functions $\alpha$ when only $ReB^2$ is fitted, colored according to the value of the corresponding dynamo number.
• Figure 4: Averaged profile of the last 24 solar cycles (black line) and region of two standard deviation from the average (gray area).
• Figure 5: Panel A: variability of the fitted models (orange area) compared to mean cycle profiles (black line). Panel B: last cycle cropped from the upper panel. Panels C and D: variability of the fitted dynamo numbers shown in the complex plane (panel C) and variability of the corresponding functions $\alpha$ (panel D). Similar colors in panels C and D indicate corresponding pairs of parameters of the fitted models. Dashed gray line in panel D shows the maximal value of cyclic variation of $|ReB|$.