In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed 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 multi-scale 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 learning-based methods with respect to the accuracy, the efficiency and the versatility.
翻译:近年来,深层学习技术被用于解决部分差异方程式(PDEs),其中物理知情神经网络(PINNs)是解决前向和反向PDE问题的一个很有希望的方法。在治理方程式中带有点源的点源代码函数(Dirac delta函数)是许多物理过程的数学模型。然而,由于Dirac delta函数带来的单一性,无法直接通过常规的PINNs方法解决。我们提出了一个以三种新颖技术解决这一问题的普遍解决方案。首先,Dirac 三角洲函数以连续概率密度函数为模型,以消除单一性;其次,建议采用一个较窄的受限制的不确定性加权算法,以平衡点源区域与其它区域之间的 PINNs损失;第三,使用具有定期激活功能的多尺度深神经网络来提高PINNs方法的精确度和趋同速度。我们用三种具有代表性的 PDEs 和实验结果来评估拟议方法,显示我们的方法在准确性、效率和反向方面超越了现有的深学习方法。