随着人工智能和深度学习的高速发展，利用机器学习方法解决科学领域的诸多问题正逐渐成为科学研究的新范式。在流体力学领域，基于深度学习算法对大量的实验测量数据和数值模拟数据进行学习，挖掘数据的内在联系，已被广泛应用于气动外形优化和湍流建模等研究。然而，经典深度学习算法框架更适用于在有大量离线数据集下的高效学习，当训练集中仅包含流场的部分信息时，经典深度学习算法通常无法有效发挥其非线性拟合能力。融合物理神经网络 (Physics informed neural networks, PINNs) 在算法设计中同时引入了数据的拟合偏差和偏微分方程的方程残差，通过对数据偏差和方程残差的同时训练，PINNs在经典深度学习算法的基础上给出了小数据集下的有效训练方案，为机器学习算法和偏微分方程求解的结合提供了新的思路。面向流体实验测量中常出现的稀疏测量情形，本文对PINNs方法应用于流场中的稀疏数据挖掘问题分别做了应用研究和算法研究。

应用研究方面，针对实验数据的有缺陷性，本文基于PINNs方法在小数据集下的灵活训练能力，对PINNs方法应用于稀疏/缺失数据下的流场重构问题进行了定性和定量研究，以层流状态下的圆柱绕流为例，详细分析了不同稀疏程度和不同缺失程度数据作为训练集时采用PINNs方法进行流场重构的准确度，评估了PINNs方法增强现有实验测量手段的可行性。

算法研究方面，虽然PINNs方法近年已成功应用于应用数学领域下的诸多分支，但PINNs方法在算法层面仍有待完善，因此鉴于PINNs方法在求解Navier-Stokes (NS) 方程时的低效性，本文分别从传统数值方法的角度和经典机器学习算法的角度对PINNs方法进行了改进，使其能更有效地进行稀疏数据挖掘，完成稀疏流场数据下的完整流场重构。

借鉴传统数值方法中的域分解策略，本文针对稀疏重构这一流动反问题提出了时空并行策略，以并行计算的方式高效快速求解每个子域，大幅提升用PINNs进行流场重构的训练效率，并通过分析PINNs方法在时空域上预测的精度特征引入了重叠式域分解策略，增强了子域数据通信的可靠性。同时，针对较高雷诺数的湍流问题，本文采用了在PINNs网络的损失函数中约束雷诺平均NS方程 (Reynolds-Averaged Navier-Stokes equations, RANS)的策略，在基本一致的计算资源下实现了更高的重构精度。本文提出的时空并行计算框架在湍流算例中均表现出稳定的并行性能。

借鉴经典机器学习算法中的数据标准化方法，考虑到PINNs方法中各输入输出特征具有底层物理含义，本文创新性地提出基于有限训练集中的样本信息同时对数据和方程做耦合式的线性变换，使得标准化后的数据满足标准化后的方程，提出了真正适用于PINNs的标准化方法。测试算例的结果表明，采用标准化方法，可以在不引入任何额外计算资源的前提下获得更高的重构精度。

With the rapid development of artificial intelligence and deep learning, using machine learning methods to solve various problems in the scientific field is gradually becoming a new paradigm for scientific research. In the field of fluid mechanics, based on deep learning algorithms, learning from a large amount of experimental measurement data and numerical simulation data to explore the inherent connections in the data has been widely used in studies such as aerodynamic shape optimization and turbulence modeling. However, classical deep learning algorithm frameworks are more suitable for efficient learning with a large offline dataset. When the training set only contains partial information of the flow field, classical deep learning algorithms often fail to effectively leverage their nonlinear fitting capabilities. Physics Informed Neural Networks (PINNs) integrate fitting biases of the data and residual of partial differential equations in algorithm design. By simultaneously training on data biases and equation residuals, PINNs provide an effective training solution for small datasets, building on classical deep learning algorithms and offering new ideas for the combination of machine learning algorithms and partial differential equation solving. This paper focuses on the application and algorithmic research of PINNs for sparse data mining problems in fluid flow measurements.

In terms of application research, addressing the imperfections of experimental data, this paper qualitatively and quantitatively studies the application of PINNs for flow reconstruction problems under sparse/incomplete data, using the flexibility of PINNs in training with small datasets. Taking the flow past a circular cylinder in laminar flow state as an example, the paper analyzes in detail the accuracy of flow reconstruction using the PINNs method when different levels of sparsity and incomplete data are used as training sets, evaluating the feasibility of enhancing existing experimental measurement methods with the PINNs method.

In terms of algorithmic research, although the PINNs method has been successfully applied in many branches of applied mathematics in recent years, there is still room for improvement at the algorithmic level. Therefore, given the inefficiency of the PINNs method in solving the Navier-Stokes (NS) equations, this paper proposes improvements to the PINNs method from the perspectives of traditional numerical methods and classical machine learning algorithms, allowing it to more effectively conduct sparse data mining and complete full flow reconstruction under sparse flow field data.

Drawing inspiration from the domain decomposition strategy in traditional numerical methods, this paper proposes a spatiotemporal parallel strategy for the inverse problem of sparse reconstruction in fluid dynamics, efficiently solving each subdomain in a parallel computing manner to significantly enhance the training efficiency of flow field reconstruction with PINNs. The paper introduces an overlapping domain decomposition strategy based on the accuracy characteristics predicted by PINNs in the spatiotemporal domain to enhance the reliability of subdomain data communication. Additionally, for turbulent problems at higher Reynolds numbers, this paper introduces constraints of the Reynolds-Averaged Navier-Stokes equations (RANS) in the loss function of the PINNs network, achieving higher reconstruction accuracy with essentially consistent computational resources. The spatiotemporal parallel computing framework proposed in this paper demonstrates stable parallel performance in turbulent flow cases.

Inspired by the data normalization methods in classical machine learning algorithms, considering that each input-output feature in the PINNs method carries underlying physical meanings, this paper innovatively proposes a coupled linear transformation based on sample information in the finite training set to simultaneously apply normalization to data and equations. This method ensures that the normalization data meet the standardized equations, presenting a normalization method truly suitable for PINNs. Results from test cases indicate that using the normalization method can achieve higher reconstruction accuracy without introducing any additional computational resources.