<p>Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool in scientific machine learning for solving differential equations. This paper presents a PINNs-based methodology for addressing singular boundary value problems (SBVPs) involving Lane-Emden type equations, which are prevalent in astrophysics and other scientific domains. The proposed approach integrates the governing differential equations and boundary conditions directly into the neural network architecture through a composite loss function, thereby eliminating the need for traditional mesh-based discretization. We demonstrate the effectiveness of our method through comprehensive numerical experiments on both linear and nonlinear SBVPs, including thermal-distribution problems with unknown analytical solution. A detailed sensitivity analysis examined the impact of key hyperparameters, such as activation function, learning rate, loss weight configuration, and network architecture. The results show that the proposed PINN framework achieves an excellent agreement with the exact solution. This study highlights the advantages of PINNs in solving challenging SBVPs, including their meshless nature and capacity to incorporate physical constraints directly into the learning process.</p>

错误:搜索内容不能为空,请输入英文关键词
错误:关键词超出字数限制,请精简
高级检索

Physics-Informed Neural Networks: A deep learning tool for solving singular boundary value problems of Lane-Emden type

  • Saurabh Tomar

摘要

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool in scientific machine learning for solving differential equations. This paper presents a PINNs-based methodology for addressing singular boundary value problems (SBVPs) involving Lane-Emden type equations, which are prevalent in astrophysics and other scientific domains. The proposed approach integrates the governing differential equations and boundary conditions directly into the neural network architecture through a composite loss function, thereby eliminating the need for traditional mesh-based discretization. We demonstrate the effectiveness of our method through comprehensive numerical experiments on both linear and nonlinear SBVPs, including thermal-distribution problems with unknown analytical solution. A detailed sensitivity analysis examined the impact of key hyperparameters, such as activation function, learning rate, loss weight configuration, and network architecture. The results show that the proposed PINN framework achieves an excellent agreement with the exact solution. This study highlights the advantages of PINNs in solving challenging SBVPs, including their meshless nature and capacity to incorporate physical constraints directly into the learning process.