Data-driven discovery of dynamo cycle equations

TL;DR

This paper presents a data-driven framework combining DMD and SINDy to model stellar dynamo cycles directly from numerical data, outperforming traditional methods in robustness and applicability.

Contribution

The authors develop a novel approach using DMD and SINDy to recover and predict stellar dynamo equations from data, including regimes where classical analysis fails.

Findings

SINDy-derived models are more robust than WNL analysis.

Models can predict magnetic saturation amplitudes.

Applicable to nonlinear regimes where WNL cannot be used.

Abstract

Many low-mass stars like the Sun host periodic, oscillatory magnetic fields that lead to variable levels of stellar activity, driving space weather that affects the habitability and detection of exoplanets. Owing to the intrinsic difficulty in modeling stellar magnetohydrodynamics across scales, realistic numerical simulations of this process are very challenging, and developing reduced-order models is of interest. In this work, we develop a framework to recover such models directly from numerical data by using a combination of Dynamic Mode Decomposition (DMD) to identify coherent magnetic structures, and the Sparse Identification of Nonlinear Dynamics (SINDy) framework to model their dynamics. We compare these models to those obtained using the classic mathematical method of weakly nonlinear (WNL) analysis. This approach is implemented on a one-dimensional mean-field dynamo model that parameterizes the main components of a convective dynamo in a low-mass star -- helical convection and differential rotation. We recover oscillatory dynamo models as a function of the dynamo strength parameter $D\sim \alpha \Omega'$, magnetic dissipation parameter $\kappa$, and a comprehensive dynamo model that predicts the magnetic state for a combination of these two parameters. Our results suggest that equations discovered with SINDy are more robust than equations from WNL analysis, and can predict the saturation amplitude of magnetic fields in parameter regimes far from the onset of dynamo action characterized by stiff nonlinearities. This includes unstable, and typically unknown, subcritical branches. Further to this, SINDy is able to find equations in parameter regimes where the nonlinearity is not analytic and WNL analysis cannot be applied. These properties of data-driven SINDy models suggest them as a viable alternative for modeling of stellar dynamo cycles directly from the data.