Unsupervised Learning Method for the Wave Equation Based on Finite
  Difference Residual Constraints Loss
Xin Feng Yi Jiang Jia-Xian Qin Lai-Ping Zhang Xiao-Gang Deng
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.