Variational autoencoder (VAE) architectures have the potential to develop reduced-order models (ROMs) for chaotic fluid flows. We propose a method for learning compact and near-orthogonal ROMs using a combination of a $\beta$-VAE and a transformer, tested on numerical data from a two-dimensional viscous flow in both periodic and chaotic regimes. The $\beta$-VAE is trained to learn a compact latent representation of the flow velocity, and the transformer is trained to predict the temporal dynamics in latent space. Using the $\beta$-VAE to learn disentangled representations in latent-space, we obtain a more interpretable flow model with features that resemble those observed in the proper orthogonal decomposition, but with a more efficient representation. Using Poincar\'e maps, the results show that our method can capture the underlying dynamics of the flow outperforming other prediction models. The proposed method has potential applications in other fields such as weather forecasting, structural dynamics or biomedical engineering.

翻译：变分自编码器架构具有为混沌流体流动开发降阶模型的潜力。我们提出了一种结合$β$-VAE和Transformer学习紧凑近正交降阶模型的方法，并在周期性及混沌状态下的二维粘性流数值数据上进行了测试。其中，$β$-VAE被训练用于学习流动速度的紧凑潜在表征，而Transformer则被训练用于预测潜在空间中的时间动态。通过利用$β$-VAE在潜在空间中学习解耦表征，我们获得了更具可解释性的流动模型，其特征类似于本征正交分解中的观测结果，但表征效率更高。基于庞加莱映射的结果表明，我们的方法能够捕捉流动的底层动态，且性能优于其他预测模型。该方法在天气预报、结构动力学或生物医学工程等领域具有潜在应用价值。