Neural differential equations for the solar dynamo

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 α effect governing regeneration of the magnetic field. We propose an approach, in which the functional form of the α 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 α-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 α-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 prospect for data-driven investigation of the closure problem in dynamo theory.