Nonlinear reduced-order modeling of compressible flow fields using deep learning and manifold learning

This paper presents a nonlinear reduced-order modeling (ROM) framework that leverages deep learning and manifold learning to predict compressible flow fields with complex nonlinear features, including shock waves. The proposed DeepManifold (DM)-ROM methodology is computationally efficient, avoids pixelation or interpolation of flow field data, and is adaptable to various grids and geometries. The framework consists of four main steps: First, a convolutional neural network-based parameterization network extracts nonlinear shape modes directly from aerodynamic geometries. Next, manifold learning is applied to reduce the dimensionality of the high-fidelity output flow fields. A multilayer perceptron-based regression network is then trained to map the nonlinear input and output modes. Finally, a back-mapping process reconstructs the full flow field from the predicted low-dimensional output modes. DM-ROM is rigorously tested on a transonic RAE2822 airfoil test case, which includes shock waves of varying strengths and locations. Metrics are introduced to quantify the model's accuracy in predicting shock wave strength and location. The results demonstrate that DM-ROM achieves a field prediction error of approximately 3.5% and significantly outperforms reference ROM techniques, such as proper orthogonal decomposition (POD)-ROM and isometric mapping (ISOMAP)-ROM, for various training sample sizes.