基于傅里叶神经算子的三维湍流大涡模拟研究

大多数神经网络主要专注于学习有限维欧几里德空间之间的映射关系，在特定控制方程下表现优异。然而，一旦参数或初始、边界条件发生变动，它们的泛化能力往往大受影响。最近，傅里叶神经算子（FNO）模型受到了广泛关注。 FNO模型具备学习无限维函数空间之间映射关系的能力，通过数据驱动的方式学习到整个偏微分方程（PDEs）家族的演化规律，因此展现出更强大的泛化能力。尽管FNO 模型在一维和二维 PDEs 求解中取得了显著成果，但在三维 PDEs 问题的研究上仍显不足。因此，准确、稳定且高效地模拟三维湍流中的非线性相互作用具有重要意义。
（1）本文首次尝试将 FNO 模型应用于三维湍流大涡模拟研究。利用高分辨率直接数值模拟（DNS）生成的湍流数据，经滤波处理后得到低分辨率滤波速度场数据，用于模型的训练和测试。在训练过程中，模型以前几个时间节点的滤波速度场作为输入，以预测下一时间节点的滤波速度场。在后验分析中，与传统亚格子模型如动态 Smagorinsky 模型（DSM）和动态混合模型（DMM）相比， FNO 模型展现出了在预测速度谱、涡量、速度增量的概率密度函数（PDFs）等统计量以及瞬时空间结构方面的优越性。此外， FNO 模型还大幅提升了计算效率，显著降低了计算成本。经过低泰勒雷诺数流场数据训练的模型，无需任何调整或重新训练，便能直接应用于更高泰勒雷诺数湍流的精准预测，显示出其出色的泛化能力。
（2）尽管 FNO 模型在短期预测中展现出较高的精度，但在长时间预测湍流时，误差累积效应逐渐显现，预测的稳定性仍有待提高。因此，我们进一步提出了隐式 U-Net 增强的傅里叶神经算子（IU-FNO），用于稳定、高效地预测湍流的长期动力学演化过程。 IU-FNO 模型采用隐式循环傅里叶层来扩展网络深度，并融入 U-Net 网络以准确预测小尺度流场结构。为了全面评估各模型的性能，我们在稳态均匀各向同性湍流、自由剪切湍流混合层以及衰减湍流中，对基于 FNO 的改进模型以及传统亚格子模型进行了系统的测试。后验分析结果显示，我们提出的IU-FNO 模型在预测湍流的速度谱、速度增量 PDFs、速度结构函数等统计量以及流场的瞬时空间结构方面比传统亚格子模型和其他基于 FNO 的改进模型都更准确。更重要的是， IU-FNO 模型展现出了长期稳定且精准的预测能力，这是其他 FNO 改进模型所未能达到的。与原始 FNO 模型相比， IU­FNO 不仅将网络参数量降低了约 75%，计算效率也远高于传统亚格子模型。此外， IU-FNO 模型对高泰勒雷诺数的泛化能力更强，并能准确地预测衰减湍流中未见过的流动状态。因此，IU-FNO模型为使用数据驱动的方法来研究复杂湍流的长期动力学问题提供了一个重要的参考。

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