Resolving Turbulent Magnetohydrodynamics: A Hybrid Operator-Diffusion Framework

Abstract: We present a hybrid machine learning framework that combines Physics-Informed Neural Operators (PINOs) with score-based generative diffusion models to simulate the full spatio-temporal evolution of two-dimensional, incompressible, resistive magnetohydrodynamic (MHD) turbulence across a broad range of Reynolds numbers ($\mathrm{Re}$). The framework leverages the equation-constrained generalization capabilities of PINOs to predict coherent, low-frequency dynamics, while a conditional diffusion model stochastically corrects high-frequency residuals, enabling accurate modeling of fully developed turbulence. Trained on a comprehensive ensemble of high-fidelity simulations with $\mathrm{Re} \in {100, 250, 500, 750, 1000, 3000, 10000}$, the approach achieves state-of-the-art accuracy in regimes previously inaccessible to deterministic surrogates. At $\mathrm{Re}=1000$ and $3000$, the model faithfully reconstructs the full spectral energy distributions of both velocity and magnetic fields late into the simulation, capturing non-Gaussian statistics, intermittent structures, and cross-field correlations with high fidelity. At extreme turbulence levels ($\mathrm{Re}=10000$), it remains the first surrogate capable of recovering the high-wavenumber evolution of the magnetic field, preserving large-scale morphology and enabling statistically meaningful predictions.