标题： 基于偏微分方程的机器学习流体力学建模数据前向生成方法 显示英文标题

标题： Method of data forward generation with partial differential equations for machine learning modeling in fluid mechanics

摘要： 人工智能（AI）在流体力学中的应用已成为一个有吸引力的课题。高保真数据是人工智能在流体力学中成功应用的关键问题之一，然而，它获取成本高昂甚至无法获得。本研究提出了一种从偏微分方程（PDEs）中高效生成数据的方法。具体来说，首先根据随机场（例如高斯随机场，GRF，计算复杂度 O(NlogN)，N 是空间点的数量）或物理定律（例如某种谱，计算复杂度 O(NM)，M 是模式数）生成 PDEs 的解，然后计算源项、边界条件和初始条件以满足 PDEs。因此，可以构建源项、边界条件和初始条件与相应 PDEs 解的数据对。分别提出了嵌入投影方法中的泊松神经网络（Poisson-NN）和嵌入多网格数值模拟中的小波变换卷积神经网络（WTCNN），用于求解不可压缩纳维-斯托克斯方程。验证了生成的数据在训练 Poisson-NN 和 WTCNN 中的可行性。结果表明，即使没有任何 DNS 数据，生成的数据也可以训练这两个模型，具有出色的泛化能力和准确性。遵循物理定律生成的数据比遵循 GRF 生成的数据在收敛速度、泛化能力和准确性方面有显著提升。 显示英文摘要

摘要： 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.