面向流体力学中机器学习建模的偏微分方程数据正向生成方法
Method of data forward generation with partial differential equations for machine learning modeling in fluid mechanics
AI总结：
针对流体力学AI应用中高保真数据获取难的问题，提出基于偏微分方程的高效数据正向生成方法，构建两类求解不可压缩Navier-Stokes方程的神经网络，验证了生成数据的训练有效性，物理规律生成的数据效果更优。
摘要：
人工智能（AI）在流体力学领域的应用已成为热门研究方向。高保真数据是AI在流体力学中成功落地的最关键要素之一，但其获取成本高昂，甚至根本无法获取。本研究提出了一种基于偏微分方程（PDEs）的高效数据正向生成方法。具体而言，首先通过随机场（例如高斯随机场GRF，计算复杂度为O(NlogN)，其中N为空间点数量）或物理规律（例如某类谱方法，计算复杂度为O(NM)，其中M为模态数量）生成PDEs的解，随后计算满足PDEs的源项、边界条件与初始条件，由此构建源项、边界条件、初始条件与对应PDEs解的数据对。本研究分别提出了嵌入投影法的泊松神经网络（Poisson-NN），以及嵌入多重网格数值模拟的小波变换卷积神经网络（WTCNN），用于求解不可压缩Navier-Stokes方程，并验证了生成数据用于训练Poisson-NN与WTCNN的可行性。结果表明，即便不使用任何直接数值模拟（DNS）数据，生成的数据也能训练这两个模型，使其具备出色的泛化能力与精度；遵循物理规律生成的数据，相比GRF生成的数据能显著提升收敛速度、泛化能力与精度。
英文摘要：
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.
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。
