用偏微分方程生成流体智能模型训练数据，无需高精度实测数据即可高效建模。
Method of data forward generation with partial differential equations for machine learning modeling in fluid mechanics
- 从PDE解出发，逆推源项、初边值条件生成数据对。
- 生成数据可使神经网络在无直接模拟数据下仍达高精度与强泛化能力。
- 按物理规律生成的数据显著提升收敛速度与模型性能，适合流体建模研究者。
人工智能在流体力学中的应用面临高保真数据获取成本高或不可得的瓶颈。本文提出一种基于偏微分方程（PDE）的高效数据前向生成方法：先生成PDE解，采用随机场（如高斯随机场，计算复杂度O(NlogN)，N为空间点数）或物理规律（如特定谱分布，复杂度O(NM)，M为模态数），再反推满足PDE的源项、边界条件和初始条件，从而构建输入-输出数据对。针对不可压缩纳维-斯托克斯方程，分别提出嵌入投影法的泊松神经网络（Poisson-NN）和嵌入多网格数值模拟的小波变换卷积神经网络（WTCNN）。验证表明，即使不使用任何直接数值模拟（DNS）数据，仅用生成数据即可训练出具有优异泛化性与准确性的模型。按物理规律生成的数据相比随机场生成方式，在收敛速度、泛化能力和精度上均有显著提升。
原文摘要 · Abstract (English)
Artificial intelligence (AI) for fluid mechanics has become attractive topic. High-fidelity data is one of most critical issues for the successful applications of AI in fluid mechanics, however, it is expensively obtained or even inaccessible. This study proposes a high-efficient data forward generation method from the partial differential equations (PDEs). Specifically, the solutions of the PDEs are first generated either following a random field (e.g. Gaussian random field, GRF, computational complexity O(NlogN), N is the number of spatial points) or physical laws (e.g. a kind of spectra, computational complexity O(NM), M is the number of modes), then the source terms, boundary conditions and initial conditions are computed to satisfy PDEs. Thus, the data pairs of source terms, boundary conditions and initial conditions with corresponding solutions of PDEs can be constructed. A Poisson neural network (Poisson-NN) embedded in projection method and a wavelet transform convolutional neuro network (WTCNN) embedded in multigrid numerical simulation for solving incompressible Navier-Stokes equations is respectively proposed. The feasibility of generated data for training Poisson-NN and WTCNN is validated. The results indicate that even without any DNS data, the generated data can train these two models with excellent generalization and accuracy. The data following physical laws can significantly improve the convergence rate, generalization and accuracy than that generated following GRF.
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