自适应损失平衡物理信息神经网络算法求解几类偏微分方程的数值解

Self-Adaptive Loss Balanced Physics Informed Neural Networks Algorithmfor Solving Numerical Solutions of Several Classes of Partial Differential Equations

摘要: 物理信息神经网络(PINN)作为将物理定律与深度学习相结合的新型计算方法，受到广泛关注。物理信息神经网络的应用已初显成效，但如何提高其有限的精度仍是PINN面临的重大挑战。自适应损失平衡物理信息神经网络(lbPINN)通过建立高斯概率模型，为每个损失项设置自适应权重来定义自适应损失函数，解决了传统物理信息神经网络在求解偏微分方程时面临的损失函数权重分配难题。本研究通过应用PINN和lbPINN两种算法系统求解一维热传导方程、二维泊松方程、一维线性对流方程及一维电报方程，数值实验表明lbPINN相比于经典的PINN表现出更优的性能，相对L2误差有不同程度的降低，证实了lbPINN在求解这几类偏微分方程中表现出更高的精确度和有效性。

Abstract: Physics-Informed Neural Networks (PINN), as a novel computational method integrating physical laws with deep learning, have garnered widespread attention. While PINN have demonstrated initial successes in applications, improving their limited accuracy remains a significant challenge. The Self-adaptive Loss Balanced Physics-Informed Neural Networks (lbPINN) addresses the difficulty of loss function weight allocation in solving partial differential equations faced by traditional PINN by establishing a Gaussian probability model and assigning adaptive weights to each loss term, thereby defining an adaptive loss function. In this study, both PINN and lbPINN algorithms were systematically applied to solve the one-dimensional heat equation, and one-dimensional telegraph equation. Numerical experiments show that, compared to the classical PINN, lbPINN exhibits superior performance, with varying degrees of reduction in relative L2 errors. This confirms that lbPINN demonstrates higher accuracy and effectiveness in solving these classes of partial differential equations.