【摘要】 为了解决传统数值方法的不足，使用神经网络求解偏微分方程成为了数值计算领域的研究热点。以二维稳态热传导方程为例，尝试采用基于无监督学习的传统物理信息神经网络（physics-informed neural network, PINN）方法和自适应激活函数PINN方法求解偏微分方程，讨论采样点选取以及网络结构对PINN方法求解精度的影响。结果表明，PINN可以实现二维稳态热传导方程的无监督学习；在传统PINN方法中引入自适应激活函数加快了二维稳态热传导方程求解的速度；采样点选取相较于网络结构对PINN方法求解精度影响更大。因此，自适应激活函数PINN方法相较于传统PINN方法具有速度更快精度更高的优势。更多还原

【Abstract】 To address the limitations of traditional numerical methods, using neural networks to solve partial differential equations has become a research hotspot in numerical computation. Taking the two-dimensional steady-state heat conduction equation as an example, this study attempts to solve the partial differential equation by using both the traditional physics-informed neural network(PINN) method based on unsupervised learning and the adaptive activation function PINN method, in addition, discuss the influence of the selection on sampling point and network structure on the accuracy of the PINN method. The results indicate that PINN can achieve unsupervised learning for the two-dimensional steady-state heat conduction equation; additionally, incorporating adaptive activation functions into the traditional PINN method can accelerate the solution process. Furthermore, compared with the network structure, the choice of sampling points has a greater impact on the solving accuracy of the PINN method. Therefore, compared with the traditional PINN method, the adaptive activation function PINN method offers advantages of faster speed and higher accuracy.更多还原