Authors: Artin Khaleghi (New Jersey Institute of Technology), Qin Li (New Jersey Institute of Technology), Ziyang Zhang (New Jersey Institute of Technology), Bo Shen (New Jersey Institute of Technology), Haimin Wang (New Jersey Institute of Technology)

Modeling magnetohydrodynamic (MHD) processes such as magnetic reconnection, shock propagation, and current-sheet dynamics is essential for understanding solar flares, coronal mass ejections, and heliospheric energy transport. Traditional MHD solvers require structured meshes, CFL-limited time-stepping, and numerical Riemann solvers constraints that introduce artificial diffusion near current sheets and limit applicability to the geometrically complex, observationally sparse solar corona.

We present the first fully mesh-free, data-free framework for solving the compressible MHD equations using Physics-Informed Neural Networks (PINNs). The complete conservation laws such as continuity, momentum, magnetic induction, and total energy together with the divergence free constraint, are enforced directly as training residuals with no reliance on ground-truth solutions or interior data points. Three key innovations drive stable training: (1) a hard initial condition constraint via output transform guaranteeing exact IC satisfaction at t = 0; (2) causal loss reweighting with adaptive scheduling to enforce proper temporal evolution of wave propagation and current-sheet dynamics; and (3) characteristic-scale-informed automatic loss weighting that balances residuals across MHD wave families with disparate speeds.

We validate the solver on two canonical benchmarks: the Orszag–Tang vortex, which tests the formation of current sheets and MHD turbulence from smooth periodic initial conditions, and the 2D MHD rotor, which tests torsional Alfvén wave launching from a rotating dense disk. In both cases the solver accurately reproduces current-sheet formation, magnetic pressure gradients, and wave propagation without numerical diffusion or mesh refinement. Its mesh-free, continuously differentiable solution representation enables analytic current density and Poynting flux diagnostics at arbitrary resolution, and its architecture naturally accommodates sparse observational constraints from SDO/HMI and Parker Solar Probe offering a new pathway toward physics-driven coronal MHD modeling.