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.

翻译：波动方程是一类重要的物理偏微分方程，近年来，深度学习方法在加速或替代传统数值求解方面展现出潜力。然而，现有深度学习方法存在数据获取成本高、训练效率低以及边界条件泛化能力不足等问题。针对这些挑战，本文提出一种基于有限差分残差约束的无监督波动方程学习方法。我们基于结构化网格与有限差分方法构建了新型有限差分残差约束及相应的无监督训练策略，使得卷积神经网络无需依赖数据即可训练，并能够预测波的传播过程。实验结果表明，与物理信息神经网络（PINNs）类物理信息约束相比，有限差分残差约束具有更易拟合、计算成本更低以及源项泛化能力更强等优势，从而使本文方法在训练效率与应用潜力方面表现更优。