[论文解读] Solving all laminar flows around airfoils all-at-once using a parametric neural network solver

Wenbo Cao, Shixiang Tang+3 more|arXiv (Cornell University)|Jan 2, 2025

Meteorological Phenomena and Simulations被引用 3

一句话总结

本论文提出一种基于面向时间步进的神经网络求解器（TSONN）的参数化神经网络求解器，并通过网格变换来求解在定义参数空间内的所有层流翼型流动，对单个情形的升力误差约为3.6%，阻力误差约为1.4%，并在代理形式下推广到约4.6%升力和1.1%阻力，训练数据来自数亿个无标签条件。

ABSTRACT

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° to 15°. 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.

研究动机与目标

• 推进一个能够一次性处理包含广泛参数变化的翼型周围层流流动的求解器。
• 在前向与参数化问题中提升基于PINN的方法的条件性与效率。
• 展示一个可扩展的代理模型，在无标签数据的情况下对未见翼型形状和流动条件实现泛化。
• 在大参数空间内量化升力/阻力的精度表现。

提出的方法

• 使用时间步进导向的神经网络（TSONN）将病态优化重新表述为一系列条件良好的子问题。
• 应用网格变换以促进解决翼型周围的类湍流层流。
• 定义一个包含来自UIUC数据库的每个翼型扰动30%的参数空间、雷诺数100–5000、攻角−5°到15°。
• 在数亿个流动条件和翼型形状上训练一个无标签数据的代理模型。
• 在前向问题上评估升力和阻力的平均相对误差，并报告参数化代理的性能。

实验结果

研究问题

• RQ1 参数化神经网络求解器是否能够一次性覆盖翼型周围的全面层流流动空间？
• RQ2 在使用带网格变换的TSONN下，前向与参数化情景的可实现精度（升力/阻力）水平如何？
• RQ3 在所定义的参数空间内，求解器在计算时间和数据需求方面的扩展性如何？
• RQ4 代理模型在无标签数据的情况下对未见翼型形状和流动条件的泛化能力如何？

主要发现

• 基础求解器的升力系数平均相对误差约为3.6%，阻力系数约为1.4%。
• 参数化扩展几乎覆盖了定义空间内的所有层流流动情景。
• 培训涉及数亿个流动条件和翼型形状，得到的代理模型对升力误差平均为4.6%，对阻力误差为1.1%。
• 在参数空间内的所有层流流动在4.6天内求解，成本约为单次流动的40倍。
• 代理模型在高维参数问题上对无标签数据显示出强泛化性和成本效益。