Physics-constrained deep learning for solving seepage equation

With the development of deep learning algorithms, neural networks (NNs) have been widely used in physics, computer science, and other fields. Solving partial differential equations (PDEs) based on NNs is one of the research hotspots. The existing methods can be divided into two categories: one is data-driven method, and the other is physics-constrained method. Physical-constrained method is more popular due to the convenience of constructing NNs and excellent generalization ability. However, the physical-constrained method cannot ensure the solution precision when PDEs are more complex, such as seepage equations with source and sink terms. In this paper, an improved physics-constrained PDE solution method is proposed that incorporates potential features of the PDE in the loss functions. A gradient model is proposed to describe the potential feature based on spatial pressure distribution, which can be used as the additional signpost to guide NNs to approximate PDEs. The gradient models are described as special neurons that are added into the hidden layer of the NN. Based on the new ideas, the seepage equation is well solved without using any exact solutions. The effectiveness of the proposed method is verified by numerical simulation results based on PEBI grid.

Automated summary

This paper introduces an improved physics-constrained PDE solution method, which incorporates gradient models to enhance the accuracy of neural networks in approximating PDEs. This approach solves complex partial differential equation problems, such as those with source and sink terms, by adding special neurons in the hidden layer.