Data-driven symbolic expression of magnetosphere convection from Van Allen Probes

Large-scale solar-wind-driven magnetospheric convection governs the formation of global electric currents responsible for geomagnetic activity and indices. Despite its importance for space-weather dynamics, quantitative descriptions of the magnetospheric convection electric field remain limited. Widely used analytical models, such as the Volland--Stern formulation, have not been systematically constrained by in situ observations. Here, we derive closed-form symbolic expressions for the magnetospheric convection electric field directly from Van Allen Probes measurements using PhyE2E, a neural-symbolic regression framework for physics discovery. Without assuming a predefined functional form, PhyE2E decomposes the regression problem using second-order neural derivatives, synthesizes candidate symbolic expressions, and refines them through Monte Carlo tree search and genetic programming. Applied to statistical observations spanning multiple geomagnetic activity levels, the resulting symbolic formulas reproduce the observed convection electric fields with substantially improved accuracy compared with the classical Volland--Stern model. These results provide an explicit, data-driven model of inner-magnetospheric convection electric fields for space-weather studies.