We use a data-driven approach to model a three-dimensional turbulent flow
using cutting-edge Deep Learning techniques. The deep learning framework
incorporates physical constraints on the flow, such as preserving
incompressibility and global statistical invariants of velocity gradient
tensor. The accuracy of the model is assessed using statistical and
physics-based metrics. The data set comes from Direct Numerical Simulation of
an incompressible, statistically stationary, isotropic turbulent flow in a
cubic box. Since the size of the dataset is memory intensive, we first generate
a low-dimensional representation of the velocity data, and then pass it to a
sequence prediction network that learns the spatial and temporal correlations
of the underlying data. The dimensionality reduction is performed via
extraction using Vector-Quantized Autoencoder (VQ-AE), which learns the
discrete latent variables. For the sequence forecasting, the idea of
Transformer architecture from natural language processing is used, and its
performance compared against more standard Recurrent Networks (such as
Convolutional LSTM). These architectures are designed and trained to perform a
sequence to sequence multi-class classification task in which they take an
input sequence with a fixed length (k) and predict a sequence with a fixed
length (p), representing the future time instants of the flow. Our results for
the short-term predictions show that the accuracy of results for both models
deteriorates across predicted snapshots due to autoregressive nature of the
predictions. Based on our diagnostics tests, the trained Conv-Transformer model
outperforms the Conv-LSTM one and can accurately, both quantitatively and
qualitatively, retain the large scales and capture well the inertial scales of
flow but fails at recovering the small and intermittent fluid motions.

本文使用数据驱动的方法，应用前沿的深度学习技术模拟三维湍流，考虑流体的物理约束，并使用向量量化自编码器等方法降低数据维度，应用 Transformer 网络预测流体运动趋势，所得结果优于其他网络，但不能完美还原小幅动态。