Xiaogang Deng

Yiye Zou, Tianyu Li, Lin Lu et al.

Advances in deep learning have enabled physics-informed neural networks to solve partial differential equations. Numerical differentiation using the finite-difference (FD) method is efficient in physics-constrained designs, even in parameterized settings. In traditional computational fluid dynamics(CFD), body-fitted block-structured grids are often employed for complex flow cases when obtaining FD solutions. However, convolution operators in convolutional neural networks for FD are typically limited to single-block grids. To address this issue, \blueText{graphs and graph networks are used} to learn flow representations across multi-block-structured grids. \blueText{A graph convolution-based FD method (GC-FDM) is proposed} to train graph networks in a label-free physics-constrained manner, enabling differentiable FD operations on unstructured graph outputs. To demonstrate model performance from single- to multi-block-structured grids, \blueText{the parameterized steady incompressible Navier-Stokes equations are solved} for a lid-driven cavity flow and the flows around single and double circular cylinder configurations. When compared to a CFD solver under various boundary conditions, the proposed method achieves a relative error in velocity field predictions on the order of $10^{-3}$. Furthermore, the proposed method reduces training costs by approximately 20\% compared to a physics-informed neural network. \blueText{To} further verify the effectiveness of GC-FDM in multi-block processing, \blueText{a 30P30N airfoil geometry is considered} and the \blueText{predicted} results are reasonable compared with those given by CFD. \blueText{Finally, the applicability of GC-FDM to three-dimensional (3D) case is tested using a 3D cavity geometry.

Xin Feng, Yi Jiang, Jia-Xian Qin et al.

The wave equation is an important physical partial differential equation, and in recent years, deep learning has shown promise in accelerating or replacing traditional numerical methods for solving it. However, existing deep learning methods suffer from high data acquisition costs, low training efficiency, and insufficient generalization capability for boundary conditions. To address these issues, this paper proposes an unsupervised learning method for the wave equation based on finite difference residual constraints. We construct a novel finite difference residual constraint based on structured grids and finite difference methods, as well as an unsupervised training strategy, enabling convolutional neural networks to train without data and predict the forward propagation process of waves. Experimental results show that finite difference residual constraints have advantages over physics-informed neural networks (PINNs) type physical information constraints, such as easier fitting, lower computational costs, and stronger source term generalization capability, making our method more efficient in training and potent in application.