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 turbulence across a broad range of Reynolds numbers ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Re</mml:mi> </mml:mrow> </mml:mrow> </mml:math> ). 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Re</mml:mi> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>100</mml:mn> <mml:mo>,</mml:mo> <mml:mn>250</mml:mn> <mml:mo>,</mml:mo> <mml:mn>500</mml:mn> <mml:mo>,</mml:mo> <mml:mn>750</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1000</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3000</mml:mn> <mml:mo>,</mml:mo> <mml:mn>10</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mn>000</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:math> , the approach achieves state-of-the-art accuracy in regimes previously inaccessible to deterministic surrogates. At <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Re</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1000</mml:mn> </mml:mrow> </mml:math> 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Re</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>10</mml:mn> <mml:mstyle scriptlevel="0"/> <mml:mn>000</mml:mn> </mml:mrow> </mml:math> ), 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.