High-fidelity direct numerical simulation of turbulent flows for most real-world applications remains an outstanding computational challenge. Several machine learning approaches have recently been proposed to alleviate the computational cost even though they become unstable or unphysical for long time predictions. We identify that the Fourier neural operator (FNO) based models combined with a partial differential equation (PDE) solver can accelerate fluid dynamic simulations and thus address computational expense of large-scale turbulence simulations. We treat the FNO model on the same footing as a PDE solver and answer important questions about the volume and temporal resolution of data required to build pre-trained models for turbulence. We also discuss the pitfalls of purely data-driven approaches that need to be avoided by the machine learning models to become viable and competitive tools for long time simulations of turbulence.

翻译：对于大多数现实应用中的湍流进行高保真度直接数值模拟，仍然是一个悬而未决的计算挑战。最近，人们提出了几种机器学习方法以减轻计算成本，尽管这些方法在长期预测中会变得不稳定或不符合物理规律。我们发现，基于傅里叶神经算子的模型与偏微分方程求解器相结合，可以加速流体动力学模拟，从而解决大规模湍流模拟的计算开销问题。我们将FNO模型置于与PDE求解器同等重要的地位，并回答了为湍流构建预训练模型所需的数据量及时间分辨率等重要问题。我们还讨论了纯数据驱动方法的缺陷，机器学习模型需要避免这些缺陷，才能成为湍流长期模拟中可行且有竞争力的工具。