[论文解读] An Efficient Deep Learning Technique for the Navier-Stokes Equations: Application to Unsteady Wake Flow Dynamics

T. P. Miyanawala, Rajeev K. Jaiman|arXiv (Cornell University)|Oct 25, 2017

Model Reduction and Neural Networks参考文献 23被引用 95

一句话总结

本文开发了基于 CNN 的深度学习模型，用于对 Navier–Stokes 方程进行降维以预测凸体的非定常尾涡力，相比全阶模拟实现近实时预测，速度显著提升。

ABSTRACT

We present an efficient deep learning technique for the model reduction of the Navier-Stokes equations for unsteady flow problems. The proposed technique relies on the Convolutional Neural Network (CNN) and the stochastic gradient descent method. Of particular interest is to predict the unsteady fluid forces for different bluff body shapes at low Reynolds number. The discrete convolution process with a nonlinear rectification is employed to approximate the mapping between the bluff-body shape and the fluid forces. The deep neural network is fed by the Euclidean distance function as the input and the target data generated by the full-order Navier-Stokes computations for primitive bluff body shapes. The convolutional networks are iteratively trained using the stochastic gradient descent method with the momentum term to predict the fluid force coefficients of different geometries and the results are compared with the full-order computations. We attempt to provide a physical analogy of the stochastic gradient method with the momentum term with the simplified form of the incompressible Navier-Stokes momentum equation. We also construct a direct relationship between the CNN-based deep learning and the Mori-Zwanzig formalism for the model reduction of a fluid dynamical system. A systematic convergence and sensitivity study is performed to identify the effective dimensions of the deep-learned CNN process such as the convolution kernel size, the number of kernels and the convolution layers. Within the error threshold, the prediction based on our deep convolutional network has a speed-up nearly four orders of magnitude compared to the full-order results and consumes an insignificant fraction of computational resources. The proposed CNN-based approximation procedure has a profound impact on the parametric design of bluff bodies and the feedback control of separated flows.

研究动机与目标

• 激发对设计与控制中非定常凸体尾涡动力学的高效预测。
• 开发基于 CNN 的数据驱动的 MOR 框架，将凸体几何映射到流体力系数。
• 建立基于 CNN 的学习与 Mori–Zwanzig 形式在降阶建模中的联系。

提出的方法

• 使用卷积神经网络（CNN），其输入通过距离函数从凸体几何中推导。
• 使用带动量的随机梯度下降，用全阶 Navier–Stokes 数据训练 CNN。
• 采用离散卷积、非线性整流（ReLU）以及可能的池化来提取局部流场特征。
• 将 CNN 的输出映射到流体力系数，并与全阶结果进行比较。
• 提供物理类比，将 CNN 的记忆核与 Navier–Stokes 动力学的降阶表征联系起来。

实验结果

研究问题

• RQ1 在低雷诺数下，基于 CNN 的深度学习模型能否准确预测不同凸体几何形状的流体力系数？
• RQ2 CNN 超参数（卷积核尺寸、卷积核数量、层数）对预测精度与收敛性的影响是什么？
• RQ3 基于 CNN 的 MOR 与 Mori-Zwanzig 形式及简化 Navier–Stokes 动力学之间的关系如何？

主要发现

• CNN 对流体力系数的预测在定义的误差阈值内与全阶结果高度吻合。
• 基于 CNN 的方法相比全阶仿真实现近四个数量级的加速。
• 该方法在参数化凸体设计和分离流的反馈控制方面显示出潜力。
• 系统性灵敏度分析确定了有效的卷积核尺寸、核数量和层数。
• 该研究将深度核学习与简化的 Navier–Stokes 动量方程以及 Mori–Zwanzig 框架联系起来。