Solving Partial Differential Equations with Point Source Based on PhysicsInformed Neural Networks
Abstract
In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physicsinformed neural networks (PINNs) 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. We propose a universal solution to tackle this problem with three novel techniques. Firstly the Dirac delta function is modeled as a continuous probability density function to eliminate the singularity; secondly a lower bound constrained uncertainty weighting algorithm is proposed to balance the PINNs losses between point source area and other areas; and thirdly a multiscale deep neural network with periodic activation function is used to improve the accuracy and convergence speed of the PINNs method. We evaluate the proposed method with three representative PDEs, and the experimental results show that our method outperforms existing deep learningbased methods with respect to the accuracy, the efficiency and the versatility.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2111.01394
 arXiv:
 arXiv:2111.01394
 Bibcode:
 2021arXiv211101394H
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Physics  Computational Physics