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Article(id=1243301631287214545, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2025.01.001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1721404800000, receivedDateStr=2024-07-20, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774355808325, onlineDateStr=2026-03-24, pubDate=1737302400000, pubDateStr=2025-01-20, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774355808325, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774355808325, creator=13701087609, updateTime=1774355808325, updator=13701087609, issue=Issue{id=1243301630683234768, tenantId=1146029695717560320, journalId=1240685776644648972, year='2025', volume='29', issue='1', pageStart='1', pageEnd='169', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=0, createTime=1774355808181, creator=13701087609, updateTime=1774355986739, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243302379672678863, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243302379672678864, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1243301630683234768, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1, endPage=11, ext={EN=ArticleExt(id=1243301631505318356, articleId=1243301631287214545, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=High resolution turbulence flow reconstruction using flow time history deep learning, columnId=1241023037940748650, journalTitle=Journal of Ship Mechanics, columnName=Hydrodynamics, runingTitle=null, highlight=null, articleAbstract=

High-resolution time variant flow field data is the key to the study of turbulence flow. Limited by measurement methods, simulation efficiency and data storage, it is still difficult to obtain high-resolution turbulent flow data directly in some circumstances. In this paper, based on the low-dimensional representation model of flow time-history data, a neural network-based feature coding prediction model and high-resolution turbulence flow reconstruction method were proposed. Firstly, a low-dimensional representation model of the turbulence flow was established based on the one-dimensional convolution networks; then, an artificial neural network model was employed to establish the mapping between the measuring point coordinates and feature coding system, and the prediction of feature coding for the unknown measuring points was realized; finally, based on feature coding, the decoder in the representation model was utilized to generate turbulence flow time history data at unknown positions. Turbulence flow with Re=2.2×104 around a square cylinder was studied, and the low dimensional representation model and flow generation model were trained and verified. The method proposed in this paper is a high-precision turbulence flow data reconstruction method which can be widely used in one-point-based sensor data processing. It is a new approach for the reconstruction of turbulence flow field time-history data.

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湍流的研究离不开高分辨率的流场数据，但受测量方法、计算效率和数据存储等多方面限制，高分辨率湍流数据的直接获取仍比较困难。本文基于流场时程数据的低维表征模型，提出基于神经网络的特征编码预测模型与高分辨率的湍流重构方法。首先，基于一维卷积方法建立湍流时程的低维表征模型；然后，基于人工神经网络模型建立测点坐标与特征编码之间的映射关系，实现未知测点的特征编码预测；最后，利用所预测的特征编码结合表征模型的解码器生成求解域内任意位置处的湍流时程。对Re=2.2×104的方柱湍流场进行低维表征，进而实现高分辨率流场时程数据的重构，并验证方法的准确性。本文所提方法是一种在时间维度上具有高精度的湍流重构方法，且是一种无监督训练方法，可广泛应用于基于一点的传感器数据处理，是一种适用于湍流流场时程数据重构的新方法。

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通讯作者，E-mail：
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战庆亮（1987-），男，博士，讲师，通讯作者，E-mail：

葛耀君（1958-），男，博士，教授。

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战庆亮（1987-），男，博士，讲师，通讯作者，E-mail：

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战庆亮（1987-），男，博士，讲师，通讯作者，E-mail：

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葛耀君（1958-），男，博士，教授。

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葛耀君（1958-），男，博士，教授。

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湍流场高分辨重构的时程深度学习方法
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战庆亮 1, 2 , 白春锦 1 , 葛耀君 2
船舶力学 | 流体力学 2025,29(1): 1-11
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船舶力学 | 流体力学 2025, 29(1): 1-11
湍流场高分辨重构的时程深度学习方法
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战庆亮1, 2 , 白春锦1, 葛耀君2
作者信息
  • 1.大连海事大学　交通运输工程学院，辽宁　大连　116026
  • 2.同济大学　桥梁结构抗风技术　交通行业重点实验室，上海　200092

    • 战庆亮（1987-），男，博士，讲师，通讯作者，E-mail：

    葛耀君（1958-），男，博士，教授。

通讯作者:

通讯作者，E-mail：
High resolution turbulence flow reconstruction using flow time history deep learning
Qing-liang ZHAN1, 2 , Chun-jin BAI1, Yao-jun GE2
Affiliations
  • 1.College of Transportation and Engineering, Dalian Maritime University, Dalian 116026, China
  • 2.Key Laboratory of Transport Industry of Wind Resistant Technology for Bridge Structures, Tongji University, Shanghai 200092, China
出版时间: 2025-01-20 doi: 10.3969/j.issn.1007-7294.2025.01.001
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摘要
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湍流的研究离不开高分辨率的流场数据，但受测量方法、计算效率和数据存储等多方面限制，高分辨率湍流数据的直接获取仍比较困难。本文基于流场时程数据的低维表征模型，提出基于神经网络的特征编码预测模型与高分辨率的湍流重构方法。首先，基于一维卷积方法建立湍流时程的低维表征模型；然后，基于人工神经网络模型建立测点坐标与特征编码之间的映射关系，实现未知测点的特征编码预测；最后，利用所预测的特征编码结合表征模型的解码器生成求解域内任意位置处的湍流时程。对Re=2.2×104的方柱湍流场进行低维表征，进而实现高分辨率流场时程数据的重构，并验证方法的准确性。本文所提方法是一种在时间维度上具有高精度的湍流重构方法，且是一种无监督训练方法，可广泛应用于基于一点的传感器数据处理，是一种适用于湍流流场时程数据重构的新方法。

湍流重构  /  湍流流场时程  /  深度学习  /  特征提取  /  无监督模型

High-resolution time variant flow field data is the key to the study of turbulence flow. Limited by measurement methods, simulation efficiency and data storage, it is still difficult to obtain high-resolution turbulent flow data directly in some circumstances. In this paper, based on the low-dimensional representation model of flow time-history data, a neural network-based feature coding prediction model and high-resolution turbulence flow reconstruction method were proposed. Firstly, a low-dimensional representation model of the turbulence flow was established based on the one-dimensional convolution networks; then, an artificial neural network model was employed to establish the mapping between the measuring point coordinates and feature coding system, and the prediction of feature coding for the unknown measuring points was realized; finally, based on feature coding, the decoder in the representation model was utilized to generate turbulence flow time history data at unknown positions. Turbulence flow with Re=2.2×104 around a square cylinder was studied, and the low dimensional representation model and flow generation model were trained and verified. The method proposed in this paper is a high-precision turbulence flow data reconstruction method which can be widely used in one-point-based sensor data processing. It is a new approach for the reconstruction of turbulence flow field time-history data.

turbulence flow reconstruction  /  turbulence flow time history  /  deep learning  /  feature extraction  /  unsupervised model
战庆亮, 白春锦, 葛耀君. 湍流场高分辨重构的时程深度学习方法. 船舶力学, 2025 , 29 (1) : 1 -11 . DOI: 10.3969/j.issn.1007-7294.2025.01.001
Qing-liang ZHAN, Chun-jin BAI, Yao-jun GE. High resolution turbulence flow reconstruction using flow time history deep learning[J]. Journal of Ship Mechanics, 2025 , 29 (1) : 1 -11 . DOI: 10.3969/j.issn.1007-7294.2025.01.001
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0　引言
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湍流场的准确数值预测是迫切需要解决的流体力学前沿问题之一，然而高雷诺数下流场中存在不同尺度的流动结构，特征极为复杂，流场的表征与重构难度较大。流场的降维模型一直是复杂流动的有效研究手段之一，可以深入分析流动的内在特征与规律。例如，本征正交分解方法[1-2]将流场数据分解为有限个非线性特征编码和对应特征值的表征，从而达到降维分析的目的。此外，动态模态分解方法[3-4]在处理时变问题方面具有更大的优势，从Koopman变换的角度将非线性的时变问题近似为线性变换过程，以此建立低维模型简化流动特征来降低分析的难度。然而这些基于矩阵分解的方法对非线性强的湍流问题研究难度较大，主要原因是采用线性变换难以完美地描述非线性湍流特征。作为处理复杂数据的高效方法，深度学习如今已经在湍流研究中得到了越来越多的应用[5]，比如高精度的湍流模型[6]、瞬态流场分解[7]、以及流动方程的求解[8]等。
另一方面，在流场的表征模型研究方面，深度学习方法也是研究的热点[9]，例如基于二维卷积神经网络对流场快照的模态分解方法[10]和非定常流动的自动编码模型[11]，这些研究使得通过部分数据或者低分辨率数据来获取时空的高分辨率结果成为可能，例如下采样多尺度模型[12]、多重时间路径卷积模型[13]等。另外，可通过低维表征模型建立少数信息点的物理信息与整场快照之间的联系，进而通过深度学习模型根据稀疏的信息点数据重构高分辨率模型[14]，例如泰森多边形辅助重构方法[15]、浅层神经网络模型[16]。同时，非定常问题的预测可以结合长短期记忆人工神经网络（LSTM）来实现[17-18]。然而，无论采用何种方法建立稀疏测点与整场数据的关联，模型在训练过程中都需要高分辨率的流场快照作为有监督学习的输入信息[19]，并建立复杂多维流场数据的低维表征。目前关于流场时程的深度学习研究较少，其中文献[20-22]提出了基于时程数据深度学习的方法对流场特征提取进行研究。文献[23]提出了基于AVC方法的流动预测方法，然而该方法对湍流的重构精度不理想且部分数据点的预测结果不稳定，限制了其在湍流高分辨率重构中的应用。
为此，本文以此为基础，建立湍流场高分辨重构的改进方法，引入神经网络预测物理空间中测点在编码空间中的坐标，提高编码和湍流时程的预测精度。以Re=2.2×104的方柱绕流场为例进行方法的验证，并开展不同样本数量对模型精度的影响研究。
1　湍流流场的低维表征模型
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与大多数基于深度学习的流动表征方法不同，本文所面向的计算对象是流场的时程数据而非瞬态快照。由于湍流流场的随机性，导致流场的时程数据具有空间、时间的双重复杂特征。即一方面在同一瞬时，受流场中不同尺度流动结构的影响，不同位置的物理量分布复杂；另一方面，观察固定位置处测点的时程信号，其时变特征也较层流流场复杂得多。
针对此特点，本文使用FTH-AE模型[23]对湍流流场时程数据实现低维度表征，编码器与解码器对湍流时程的特征提取精度是模型准确性的关键，使用可以准确提取流场时程复杂特征的一维卷积[22]作为编码器和解码器的特征提取运算方式。在此过程中输入数据经过编码器将湍流时程压缩至长度较低的特征编码，特征编码再经过解码器还原成输入数据的时程信号，其中特征编码即为流场时程数据的低维度表征，原理如图1所示。
2　基于神经网络的编码预测方法
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面向流场时程的湍流场高分辨率重构，其核心问题是准确生成未知测点处的流场时程数据。当采用FTH-AE模型的解码器进行流场重构时，难点在于准确获取未知测点对应的特征编码。文献[23]中，提出了平均向量组合法（AVC）来计算未知测点的特征编码，其本质是根据已知测点在编码空间的坐标，用线性变换的方法构造未知测点在编码空间的坐标，即
式中，λ为坐标x的测点在编码空间中的坐标，L表示该AVC方法对应的线性变换。由于编码空间是由深度学习模型在训练中获得的，因此式（1）的线性变换一般无法显示表达。这种方法对于简单的层流问题适用性较好，尤其是当已知时程的测点较多时，用线性变换构造未知测点的编码空间坐标具有较高精度。
然而在湍流问题研究过程中，发现该方法仍存在一些限制：一方面，若待求解测点附近没有已知测点或距离已知测点较远，则求解精度将受到较大影响；另一方面，由于湍流流场较复杂，因而编码器与解码器的非线性较强，使用线性变换来构造物理坐标空间与编码空间之间的关系会导致较大的误差，直接导致编码空间位置的精度不够，因此也就无法用解码器预测未知测点处的时程数据。为此，本文提出了搭建全连接网络（MLP）来建立物理坐标与特征编码之间非线性映射，原理如图2所示，可表示为
该网络的输入为样本的物理坐标x，输出为该样本的特征编码λ，f为MLP模型代表的非线性变换。选择MSE损失函数来降低残差，模型具体结构与参数见表1。其中，MLP预测编码的神经网络可以改进AVC方法的不足，其原因包括：MLP网络中包含非线性激活层，因此模型总体可具有较强的非线性，对复杂的问题适应性更强；同时当已知的数据对（坐标及其编码空间的坐标）足够多时，浅层MLP模型的训练难度较低。先获得物理空间到编码空间的非线性映射关系，实现对任意位置处时程所对应的特征编码的预测，进而使用解码器对特征编码进行解码，以此来提高对复杂流动问题的重构精度。
3　算例与分析
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本文以雷诺数Re=2.2×104的方柱绕流场为例，研究基于MLP编码预测和FTH-AE的湍流场重构的准确性。流场时程数据采用数值模拟方法获取，计算域如图3所示，整体计算域平面大小为40D×20D，其中D为方柱边长。在顺流方向，方柱中心距离上游边界10D，距离下游边界30D；在横流方向，方柱中心距离上下边界均为10D；展向厚度为10D，均匀划分30层网格。方柱近尾流区域流场变化更加复杂，故采用密集的非结构化网格，而距离方柱较远处使用稀疏非结构化网格，贴近方柱壁面处则采用结构化网格，如图3（b）所示。方柱表面采用无滑移壁面边界，入口边界条件为速度入口，U=0.33 m/s，出口边界条件为压力出口，横流向与上、下边界条件均为对称边界条件。此外，采用Wale-LES亚格子模型修正涡粘系数，更准确模拟湍流流动。
方柱壁面附近及尾流区域中的流动特征复杂，因此在其周围布置了7120个测点以获取各测点的流动信息，随机分布在流场计算网格密集区域内，测点分布情况如图4所示。采用嵌入深度学习数据输出后处理模块的zFlower程序[24]，在流场计算的每个时间步结束时，每个测点都可以输出该位置的流向速度，计算完成后得到流向速度的时程数据集合。
3.1　模型参数
流动表征模型包括编码器和解码器，其中，编码器由三层卷积层、附加Flatten层及两层全连接层组成，卷积层逐层提取数据中蕴含的抽象特征，同时压缩数据至更低的维度。考虑到本文输入数据的尺寸较大，所以选择尺寸为13的大卷积核来获得更好的局部视野，经过编码器后输入数据被压缩为长度为20的特征编码，该向量为输入数据的低维表征，可经过解码器还原为输入数据。解码器由全连接层、Reshape层与4层反卷积层组成模型参数与编码部分相对应，其中反卷积层可以看作逆向卷积，即从高度抽象的特征编码中重构出输入数据的原始时程。由此，该模型实现了对数据的低维度表征与还原。为了研究不同样本数据量情况下模型的准确性，共设置了4个算例，分别使用不同数量的时程样本实现湍流的表征与预测，如表2所示。
模型训练中使用MSE作为损失函数，模型优化器为Adam，并使用默认学习率，设置训练次数为3000，训练过程中提前结束（Earlystopping）设置为50，即一旦残差值在50步内不再下降，模型训练将自动停止。所得到的训练过程中模型损失函数变化情况如图5所示。
由图5可以看出，4个算例都取得了较高的模型精度，其中Case1和Case2的残差值较低，Case3与Case4残差值相近且均高于Case1与Case2，可见样本数量是影响模型精度的重要因素。需要指出的是，虽然Case4有着不错的残差值，但其损失曲线的波动明显偏大，主要原因为其训练集当中的测点分布过于稀疏，因而训练集无法全面覆盖各类流场特征，而模型在训练时不断调整权重导致其损失曲线的波动偏大。
3.2　FTH-AE模型表征准确性验证
FTH-AE模型训练的损失值即为输入数据与重构数据的残差，该值可以准确地反映模型精度，即损失值越小，说明网络提取的输入数据特征越精准。为了进一步验证模型的准确性，本文在4个训练集之中随机选取了6条样本来比较原始曲线与模拟曲线的差异值。由图6可见，原始数据与模型还原曲线的6组对比差异与图5中的残差值相对应，各数据集中两条曲线几乎重合，说明该FTH-AE模型可以准确地提取到流场中各个测点处数据的低维度特征。
为了全面地展示出模型对整个流场内不同流动特征的提取精度，本文使用各点位置处的输入时程与模型解码时程之间误差（relative error）Rerr作为指标[23]，将各点的误差值与物理坐标结合绘制散点图，如图7所示。其中，绿色表示误差值位于0~0.01，蓝色表示误差值位于0.01~0.05，黄色表示误差值位于0.05~0.15，橘色表示误差值位于0.15~0.3，红色表示误差值大于0.3。由图可见，本文的模型可以很好地对输入时程进行重构，说明该模型对输入数据的低维表征是准确的。
3.3　生成时程的准确性验证
根据FTH-AE模型所得到的样本特征编码及样本测点的坐标，可以使用已知的训练样本对MLP网络进行训练，训练过程的损失函数变化如图8所示。
当MLP模型误差足够小时，即可以用其对任意位置处的物理坐标进行特征编码的预测。并进一步使用FTH-AE的解码器对特征编码进行解码，得到该位置处的预测。同样，随机选取6条FTH-AE模型预测时程与输入样本进行比较，其结果如图9所示。
对比图9中的结果可发现，本文方法成功地对流场中任意位置时程进行了预测，但其中Case3与Case4的预测结果偏差较大。将各时程预测的误差列于图10，可知模型预测精度同样与模型训练的残差值相对应。对比图7与图10可以看出，图7中时程曲线的精度要高于图10中的时程曲线。原因是图7中的时程为模型的已知训练样本，而图10中时程曲线是根据特征编码预测模型与湍流时程生成模型预测得到的，即模型在训练过程中没有“见过”这个时程样本，所以图10中时程曲线的精度略低是合乎情理的。同时可以发现，相对误差较大的时程曲线大多位于上游来流区域，这是因为上游来流区域内的流场几乎没有受到柱体的扰动，使得上游区域内时程曲线近似为直线，没有明显的流动特征，导致相对误差较大，但预测的绝对误差并不大。
进一步地，本文对流场任意位置处的时程曲线进行预测并取同一瞬时结果，即可得到流场的瞬态云图，如图11所示。图中左侧一列为输入的模型训练样本散点图，右侧一列为模型预测的湍流场瞬态云图。可以看出，4种不同策略的数据集均可以重构出整个流场，其中Case1与Case2的预测结果都与真实的流场云图非常接近，而Case3的整体预测结果虽然稍差于前两个算例的结果，但也较好地预测出了流场云图，Case4的结果在部分位置出现了一定的偏差。由此可见，本文方法成功地对稀疏的输入流场进行了重构，对于不同疏密程度的数据集都取得了不错的效果，模型的训练数据样本数越多，可得到的结果就更精确；同时结果也表明，即使已知样本非常稀疏，本文的方法依然具有较好的强健性。
4　结论
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本文提出了适用于湍流场高分辨率流动重构的时程深度学习模型，获得了复杂湍流的低维表征、建立了物理坐标与特征编码之间的非线性机器学习模型，进而实现了湍流流场高分辨率重构，同时开展了训练样本数量对模型的精度影响研究，得到如下结论：
（1）本文提出的时程深度学习方法，可较准确地将全场的湍流流场时程投影到低维编码空间中，实现湍流流场的低维表征。
（2）利用时程深度学习模型结合MLP编码预测网络，可以实现计算域内任一点处流场时程的重构，并以Re=2.2×104的方柱绕流场为例进行了高分辨率的流场预测。
（3）本文采用的时程深度学习模型具有很好的强健性，同时，对于数据稀疏的已知数据集仍可以预测出较为准确的流场时程，可用于基于测点传感器的湍流场数据研究。
基金
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  • 国家自然科学基金资助项目(51778495; 51978527)
  • 桥梁结构抗风技术交通行业重点实验室（上海）开放课题(KLWRTBMC21-02)
  • 辽宁教育厅研究计划(LJKZ0052)
  • 中央高校基本科研业务费专项资金资助(3132022189)
文献
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2025年第29卷第1期
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doi: 10.3969/j.issn.1007-7294.2025.01.001
  • 接收时间：2024-07-20
  • 首发时间：2026-03-24
  • 出版时间：2025-01-20
补充材料
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  • 收稿日期：2024-07-20
基金
国家自然科学基金资助项目(51778495; 51978527)
桥梁结构抗风技术交通行业重点实验室（上海）开放课题(KLWRTBMC21-02)
辽宁教育厅研究计划(LJKZ0052)
中央高校基本科研业务费专项资金资助(3132022189)
作者信息
    1.大连海事大学　交通运输工程学院，辽宁　大连　116026
    2.同济大学　桥梁结构抗风技术　交通行业重点实验室，上海　200092

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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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