Despite some empirical successes for solving nonlinear evolution equations using deep learning, there are several unresolved issues. First, it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well. Second, the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks. In this paper, we propose a new architecture that combines the deep residual neural network with some underlying physical laws. Using the sine-Gordon equation as an example, we show that the numerical result is in good agreement with the exact soliton solution. In addition, a lot of numerical experiments show that the model is robust under small perturbations to a certain extent.

尽管在使用深度学习解决非线性演化方程方面取得了一些经验性的成功，但仍有一些未解决的问题。首先，它无法揭示某些方程式的动力学行为，其中很好地包含了高度非线性的源项。其次，对于传统的前馈神经网络，经常会出现梯度爆炸和消失的问题。在本文中，我们提出了一种将深残留神经网络与一些潜在物理定律相结合的新架构。以正弦-戈登方程为例，我们证明了数值结果与精确的孤子解吻合良好。此外，大量数值实验表明，该模型在一定程度的小扰动下具有鲁棒性。