An emerging trend in deep learning research focuses on the applications of graph neural networks (GNNs) for mesh-based continuum mechanics simulations. Most of these learning frameworks operate on graphs wherein each edge connects two nodes. Inspired by the data connectivity in the finite element method, we present a method to construct a hypergraph by connecting the nodes by elements rather than edges. A hypergraph message-passing network is defined on such a node-element hypergraph that mimics the calculation process of local stiffness matrices. We term this method a finite element-inspired hypergraph neural network, in short FEIH($φ$)-GNN. We further equip the proposed network with rotation equivariance, and explore its capability for modeling unsteady fluid flow systems. The effectiveness of the network is demonstrated on two common benchmark problems, namely the fluid flow around a circular cylinder and airfoil configurations. Stabilized and accurate temporal roll-out predictions can be obtained using the $φ$-GNN framework within the interpolation Reynolds number range. The network is also able to extrapolate moderately towards higher Reynolds number domain out of the training range.
Accurate prediction over long time horizons is crucial for modeling complex physical processes such as wave propagation. Although deep neural networks show promise for real-time forecasting, they often struggle with accumulating phase and amplitude errors as predictions extend over a long period. To address this issue, we propose a novel loss decomposition strategy that breaks down the loss into separate phase and amplitude components. This technique improves the long-term prediction accuracy of neural networks in wave propagation tasks by explicitly accounting for numerical errors, improving stability, and reducing error accumulation over extended forecasts.
In this paper, we present a physics-based deep learning framework for data-driven prediction of wave propagation in fluid media. The proposed approach, termed Multistep Integration-Inspired Attention (MI2A), combines a denoising-based convolutional autoencoder for reduced latent representation with an attention-based recurrent neural network with long-short-term memory cells for time evolution of reduced coordinates. This proposed architecture draws inspiration from classical linear multistep methods to enhance stability and long-horizon accuracy in latent-time integration. Despite the efficiency of hybrid neural architectures in modeling wave dynamics, autoregressive predictions are often prone to accumulating phase and amplitude errors over time. To mitigate this issue within the MI2A framework, we introduce a novel loss decomposition strategy that explicitly separates the training loss function into distinct phase and amplitude components. We assess the performance of MI2A against two baseline reduced-order models trained with standard mean-squared error loss: a sequence-to-sequence recurrent neural network and a variant using Luong-style attention. To demonstrate the effectiveness of the MI2A model, we consider three benchmark wave propagation problems of increasing complexity, namely one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Our results demonstrate that the MI2A framework significantly improves the accuracy and stability of long-term predictions, accurately preserving wave amplitude and phase characteristics. Compared to the standard long-short term memory and attention-based models, MI2A-based deep learning exhibits superior generalization and temporal accuracy, making it a promising tool for real-time wave modeling.
In this paper, we present a deep learning technique for data-driven predictions of wave propagation in a fluid medium. The technique relies on an attention-based convolutional recurrent autoencoder network (AB-CRAN). To construct a low-dimensional representation of wave propagation data, we employ a denoising-based convolutional autoencoder. The AB-CRAN architecture with attention-based long short-term memory cells forms our deep neural network model for the time marching of the low-dimensional features. We assess the proposed AB-CRAN framework against the standard recurrent neural network for the low-dimensional learning of wave propagation. To demonstrate the effectiveness of the AB-CRAN model, we consider three benchmark problems, namely, one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Using the spatial-temporal datasets from the benchmark problems, our novel AB-CRAN architecture accurately captures the wave amplitude and preserves the wave characteristics of the solution for long time horizons. The attention-based sequence-to-sequence network increases the time-horizon of prediction compared to the standard recurrent neural network with long short-term memory cells. The denoising autoencoder further reduces the mean squared error of prediction and improves the generalization capability in the parameter space.