Spectral domain graph convolutional deep neural network for predicting unsteady and nonlinear flows

Mode decomposition methods, such as proper orthogonal decomposition and dynamic mode decomposition (DMD), have introduced a novel data-driven approach for flow prediction. These methods aim to identify a collection of modes that capture the essential flow features. Subsequently, the flow field data are projected onto these modes to reconstruct and predict the evolution of the flow field. However, due to their inherent linearity, mode decomposition methods are limited in effectively handling unsteady and nonlinear flow exhibiting significant nonlinearities. In this study, we propose a spectral graph convolutional deep neural network (SGC-DNN). It employs the eigenvectors of the Laplacian matrix as modes to fully utilize the adjacency information within the graph structure to solve flow on an unstructured grid better. Additionally, we employ a DNN (deep neural network) to model the temporal evolution of each mode, thereby enhancing the model's adaptability to nonlinear flow fields. To evaluate the performance of our proposed SGC-DNN, we compare its prediction results with those obtained using DMD and DNN for the flow around a cylinder on unstructured grids at various Reynolds numbers (ranging from 1000 to 500 000). We also compared the predictive results of these three models for flow with complex geometries, such as the Da Vinci pipeline flow and intracranial aneurysm blood flow. The comparative analysis demonstrates that SGC-DNN outperformed the other models, yielding lower L₂ relative errors and higher R² values. These outcomes highlight the superiority of SGC-DNN in accurately predicting unsteady and nonlinear flow characterized by graph structures.