We investigate the capability of machine learning (ML) based reduced order model (ML-ROM) for two-dimensional unsteady flows around a circular cylinder at different Reynolds numbers. The present ML-ROM is constructed by two ML schemes: a convolutional neural network-based autoencoder (CNN-AE) and a long short-term memory (LSTM). The CNN-AE is utilized to map high-dimensional flow fields obtained by direct numerical simulation (DNS) into a low-dimensional latent space while keeping their spatially coherent information. The LSTM is then trained to learn the temporal evolution of the mapped latent vectors together with the information on the Reynolds number. Using the trained LSTM model, the high-dimensional dynamics of flow fields can be reproduced with the aid of the decoder part of CNN-AE, which can map the predicted low-dimensional latent vector to the high-dimensional space. We find that the flow fields generated by the present ML-ROM show statistical agreement with the reference DNS data. The dependence of the accuracy of the proposed model on the Reynolds number is also examined in detail. The present results demonstrate that the ML-ROM can reconstruct flows at the Reynolds numbers that were not used in the training process unless the flow regime drastically changes.

我们研究了基于机器学习（ML）的降阶模型（ML-ROM）的能力，该模型用于绕雷诺数不同的圆柱周围的二维非定常流动。当前的ML-ROM由两种ML方案构成：基于卷积神经网络的自动编码器（CNN-AE）和长短期记忆（LSTM）。CNN-AE用于将通过直接数值模拟（DNS）获得的高维流场映射到低维潜在空间中，同时保持其空间相关信息。然后训练LSTM来学习映射的潜在向量的时间演化以及有关雷诺数的信息。使用训练有素的LSTM模型，可以借助CNN-AE的解码器部分再现流场的高维动态，可以将预测的低维潜在矢量映射到高维空间。我们发现，由当前ML-ROM生成的流字段显示出与参考DNS数据的统计一致性。还详细研究了所提出模型的准确性对雷诺数的依赖性。目前的结果表明，ML-ROM可以以训练过程中未使用的雷诺数重建流量，除非流量方式发生了急剧变化。