标题： 使用参数化神经网络求解器一次性求解所有绕翼型的层流流动 显示英文标题

标题： Solving all laminar flows around airfoils all-at-once using a parametric neural network solver

摘要： 近年来，物理信息神经网络（PINNs）在解决由偏微分方程（PDEs）控制的正向、反向和参数化问题方面引起了越来越多的研究兴趣。尽管具有前景，但由于病态性，PINNs在许多场景中仍然面临重大挑战。面向时间步进的神经网络（TSONN）通过将病态优化问题重新表述为一系列良态子问题，大大提高了其处理复杂场景的能力。本文提出了一种基于TSONN和网格变换的新求解器，用于模拟机翼周围的层流流动，并在各种测试案例中进行了验证。具体而言，该求解器在升力系数上的平均相对误差约为3.6%，在阻力系数上的平均相对误差约为1.4%。此外，本文还将该求解器扩展到涉及流动条件和机翼形状的参数化问题，涵盖了工程中几乎所有的层流流动场景。形状参数空间定义为对UIUC机翼数据库中的每个机翼应用30%的扰动，雷诺数范围从100到5000，攻角范围从-5{\deg }到15{\deg }。参数化求解器仅需4.6天即可解决参数空间内的所有层流流动，计算成本大约是求解单个流动的40倍。模型训练涉及数亿个流动条件和机翼形状，最终得到一个具有强大泛化能力的代理模型，无需标记数据。具体而言，该代理模型在升力系数上的平均误差为4.6%，在阻力系数上的平均误差为1.1%，展示了其在解决高维参数化问题和代理建模方面的高泛化性、成本效益和效率的潜力。 显示英文摘要

摘要： Recent years have witnessed increasing research interests of physics-informed neural networks (PINNs) in solving forward, inverse, and parametric problems governed by partial differential equations (PDEs). Despite their promise, PINNs still face significant challenges in many scenarios due to ill-conditioning. Time-stepping-oriented neural network (TSONN) addresses this by reformulating the ill-conditioned optimization problem into a series of well-conditioned sub-problems, greatly improving its ability to handle complex scenarios. This paper presents a new solver for laminar flow around airfoils based on TSONN and mesh transformation, validated across various test cases. Specifically, the solver achieves mean relative errors of approximately 3.6% for lift coefficients and 1.4% for drag coefficients. Furthermore, this paper extends the solver to parametric problems involving flow conditions and airfoil shapes, covering nearly all laminar flow scenarios in engineering. The shape parameter space is defined as the union of 30% perturbations applied to each airfoil in the UIUC airfoil database, with Reynolds numbers ranging from 100 to 5000 and angles of attack spanning from -5{\deg} to 15{\deg}. The parametric solver solves all laminar flows within the parameter space in just 4.6 day, at approximately 40 times the computational cost of solving a single flow. The model training involves hundreds of millions of flow conditions and airfoil shapes, ultimately yielding a surrogate model with strong generalization capability that does not require labeled data. Specifically, the surrogate model achieves average errors of 4.6% for lift coefficients and 1.1% for drag coefficients, demonstrating its potential for high generalizability, cost-effectiveness, and efficiency in addressing high-dimensional parametric problems and surrogate modeling.