标题： 解决湍流磁流体动力学：一种混合算子扩散框架 显示英文标题

标题： Resolving Turbulent Magnetohydrodynamics: A Hybrid Operator-Diffusion Framework

摘要： 我们提出了一种混合机器学习框架，将物理信息神经算子（PINOs）与基于得分的生成扩散模型相结合，以模拟二维不可压缩电阻磁流体动力学（MHD）湍流在广泛雷诺数范围内的完整时空演化（$\mathrm{Re}$）。该框架利用PINOs的方程约束泛化能力来预测相干的低频动力学，而条件扩散模型则随机校正高频残差，从而实现对完全发展湍流的精确建模。该方法在包含$\mathrm{Re} \in \{100, 250, 500, 750, 1000, 3000, 10000\}$的高保真模拟综合集合上进行训练，在以前确定性代理模型无法触及的区域内实现了最先进的准确性。在$\mathrm{Re}=1000$和$3000$处，该模型能够忠实重建速度场和磁场的完整谱能分布，深入模拟后期，以高保真度捕捉非高斯统计特性、间歇性结构和跨场相关性。在极端湍流水平（$\mathrm{Re}=10000$）下，它仍然是第一个能够恢复磁场高波数演化的代理模型，保持大尺度形态并实现具有统计意义的预测。 显示英文摘要

摘要： 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.