物理信息神经网络求解偏微分方程实践入门：来自天体物理学与等离子体物理学的基准测试
A hands-on introduction to Physics-Informed Neural Networks for solving partial differential equations with benchmark tests taken from astrophysics and plasma physics
AI总结：
本文以实践教程形式介绍PINNs通过残差最小化及软硬边界约束求解PDE，并在Grad-Shafranov和Lane-Emden方程上验证其灵活性、参数与反问题能力，配套开源代码。
摘要：
我介绍了深度学习和神经网络在求解偏微分方程（PDEs）中的应用。该方法称为物理信息神经网络（PINNs），通过最小化在定义域内多个点上计算得到的方程残差来工作。边界条件的引入有两种方式：一是在最小化过程中加入软约束及相应的边界数据值，二是通过硬约束严格强制解满足边界条件。PINNs在从二维物理/天体物理问题中提取的多种PDE上进行了测试。具体而言，我们研究了描述磁主导等离子体中磁流体动力学平衡的类Grad-Shafranov方程，同时也考虑了模拟自引力流体静力学平衡下恒星内部结构的Lane-Emden方程。通过多个示例展示了该方法处理各类边界条件的灵活性，以及其在求解参数化问题和反问题方面的便捷性。基于PyTorch/TensorFlow库的相应Python代码已公开提供。
英文摘要：
I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the equation evaluated at various points within the domain. Boundary conditions are incorporated either by introducing soft constraints with corresponding boundary data values in the minimization process or by strictly enforcing the solution with hard constraints. PINNs are tested on diverse PDEs extracted from two-dimensional physical/astrophysical problems. Specifically, we explore Grad-Shafranov-like equations that capture magnetohydrodynamic equilibria in magnetically dominated plasmas. Lane-Emden equations that model internal structure of stars in sef-gravitating hydrostatic equilibrium are also considered. The flexibility of the method to handle various boundary conditions is illustrated through various examples, as well as its ease in solving parametric and inverse problems. The corresponding Python codes based on PyTorch/TensorFlow libraries are made available.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
