A Universal PINNs Method for Solving Partial Differential Equations with a Point Source

In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs) method emerges to be a promising method for solving both forward and inverse PDE problems. PDEs with a point source that is expressed as a Dirac delta function in the governing equations are mathematical models of many physical processes. However, they cannot be solved directly by conventional PINNs method due to the singularity brought by the Dirac delta function. In this paper, we propose a universal solution to tackle this problem by proposing three novel techniques. Firstly the Dirac delta function is modeled as a continuous probability density function to eliminate the singularity at the point source; secondly a lower bound constrained uncertainty weighting algorithm is proposed to balance the physics-informed loss terms of point source area and the remaining areas; and thirdly a multi-scale deep neural network with periodic activation function is used to improve the accuracy and convergence speed. We evaluate the proposed method with three representative PDEs, and the experimental results show that our method outperforms existing deep learning based methods with respect to the accuracy, the efficiency and the versatility.