The 3D pulsar magnetosphere with machine learning: First results

¹ Department of Physics, University of Patras Rio 26504, Greece
² Research Center for Astronomy and Applied Mathematics, Academy of Athens Athens 11527, Greece
³ Department of Electrical Engineering, National Technical University of Athens Athens 15772, Greece

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Received: 18 February 2025
Accepted: 16 November 2025

Abstract

Context. All particle-in-cell (PIC) simulations of the pulsar magnetosphere over the past decade show closed line regions that end a significant distance inside the light cylinder, and manifest thick, strongly dissipative separatrix surfaces instead of thin current sheets, with a tip that has a distinct pointed Y shape rather than a T shape.

Aims. We need to understand the origin of these results, which were not predicted by our earlier numerical simulations of the pulsar magnetosphere. To gain new insight into this problem, we set out to obtain the theoretical steady-state solution of the ideal 3D force-free magnetosphere with zero dissipation along the separatrix and equatorial current sheets. To achieve this goal, we developed a novel numerical method.

Methods. We solved two independent magnetospheric problems without current sheet discontinuities in the domains of open and closed field lines and adjusted the shape of their interface (the separatrix) to satisfy the pressure balance between the two regions. We obtained the solution using meshless physics-informed neural networks (PINNs).

Results. We present our first results for an inclined dipole rotator using the new methodology. We are able to zoom-in around the Y-point and inside the closed line region, and we observe new interesting features. This is the first time the steady-state 3D problem is addressed directly, rather than through a time-dependent simulation that eventually relaxes to a steady state.

Conclusions. We trained a neural network that instantaneously yields the three components of the magnetic field and their spatial derivatives at any given point. Our results demonstrate the potential of the new method to generate new solutions of the ideal pulsar magnetosphere.