This paper focuses on the PINNs algorithm by proposing the ALM-PINNs computational framework to solve various nonlinear partial differential equations and corresponding parameters identification problems. The numerical solutions obtained by the ALM-PINNs algorithm are compared with both the exact solutions and the numerical solutions implemented from the PINNs algorithm. This demonstrates that under the same machine learning framework (TensorFlow 2.0) and neural network architecture, the ALM-PINNs algorithm achieves higher accuracy compared to the standard PINNs algorithm. Additionally, this paper systematically analyzes the construction principles of the loss function by introducing the probability distribution of random errors as prior information, and provides a theoretical basis for algorithm improvement.

翻译：本文聚焦于PINNs算法，提出ALM-PINNs计算框架以求解各类非线性偏微分方程及相应的参数辨识问题。通过将ALM-PINNs算法获得的数值解与精确解及基于PINNs算法实现的数值解进行对比，表明在相同的机器学习框架（TensorFlow 2.0）与神经网络架构下，ALM-PINNs算法相比标准PINNs算法具有更高的精度。此外，本文通过引入随机误差的概率分布作为先验信息，系统分析了损失函数的构建原理，为算法改进提供了理论依据。