CN113343606B - Method for predicting full flow field from sparse sensor information based on compressed sensing reduced order model - Google Patents

Method for predicting full flow field from sparse sensor information based on compressed sensing reduced order model Download PDF

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CN113343606B
CN113343606B CN202110742982.1A CN202110742982A CN113343606B CN 113343606 B CN113343606 B CN 113343606B CN 202110742982 A CN202110742982 A CN 202110742982A CN 113343606 B CN113343606 B CN 113343606B
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张鑫帅
季廷炜
谢芳芳
郑鸿宇
郑耀
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Abstract

The invention provides a method for predicting an all-flow field from sparse sensor information by using a novel compressed sensing reduced-order model. The method provided by the invention comprises two stages, namely off-line training and on-line learning. In the offline learning phase, the sensor signals are first modeled by an LSTM model. And then, carrying out dimensionality reduction on the complex complete flow field by utilizing a sparsely driven DMD algorithm to obtain a low-dimensional mode of the complex complete flow field, and automatically selecting all the low-dimensional modes, so that higher reconstruction accuracy is obtained under the same DMD mode number. And then, obtaining a nonlinear relation between the sensor information and the low-dimensional representation of the flow field through DNN network training. And finally, reconstructing the full flow field information through the DNN predicted low-dimensional information. And in the on-line learning stage, the trained model is used for prediction in an actual experiment. The method is simple to implement and good in universality. The method can be widely applied to reconstruction and prediction of a complex full flow field from sparse point information, and the result shows that the method is high in prediction precision and good in robustness.

Description

Method for predicting full flow field from sparse sensor information based on compressed sensing reduced order model
Technical Field
The invention relates to a method for predicting an all-flow field from sparse sensor information based on a compressed sensing reduced order model, which can robustly and accurately predict the all-flow field state information from a sensor sparse measurement point and belongs to the field of fluid flow modeling and prediction.
Background
In the design of the aircraft, the total flow field of different aircraft shapes needs to be estimated, so that lift resistance information of the surface of the aircraft is obtained, and the shape design is optimized according to the change of the lift resistance. However, estimation of the all-state flow field is very time consuming and laborious. A rapid and accurate flow field assessment tool can greatly accelerate the design of the aircraft. Or in the active flow control of the cylinder, the flow field is changed by arranging jet air flow on the surface, so that the drag reduction effect is achieved, however, the regulation and control of the jet strategy needs to perform feedback control on the flow field estimation. If the information of the whole flow field can be quickly and accurately obtained, the flow control strategy can be accurately regulated and controlled.
At present, two methods are mainly adopted for realizing the prediction of the full flow field from a sparse measurement point: firstly, a mapping relation from a sparse measurement point to a full flow field is established through machine learning, and then a trained model is directly used for prediction in an experiment, but when the flow field range is large, the high-dimensional and nonlinear performance of the flow field is strong, a neural network model with huge parameters can be generated, so that the model not only needs huge cost for training, but also has low prediction accuracy of the obtained model. The other method is realized by combining machine learning and intrinsic orthogonal decomposition (POD) reduced models, the method reduces a high-dimensional flow field into a smaller dimension by using the reduced models, and then establishes a relation from sparse point measurement to low-dimensional representation of the flow field through a neural network. However, the mode of POD decomposition is a complex mode of multi-frequency superposition, and the phenomenon of multi-frequency superposition is not favorable for capturing the time evolution characteristic of the flow field.
Disclosure of Invention
The invention provides a novel method for predicting an all-flow field from sparse sensor information based on a compressed sensing reduced order model, aiming at the limitation of the existing method for predicting the state of the all-flow field from sparse measurement point information. The method provided by the invention comprises two stages, namely an off-line training stage and an on-line prediction stage. In the off-line training stage, firstly, the time sequence evolution process of the sensor signal is regarded as a dynamic system, the sensor signal is modeled through an LSTM model, and the output signal of the sensor at the future moment is predicted from the collected historical signals. And then, carrying out dimensionality reduction on the complex complete flow field by utilizing a sparsely driven DMD algorithm to obtain a low-dimensional DMD mode and a corresponding low-dimensional DMD dynamic coefficient of the complex complete flow field, automatically selecting all the low-dimensional modes, and searching for a DMD mode which has a large influence on the complete flow field, so that the DMD mode can obtain higher reconstruction accuracy under the same DMD mode number. And then, a nonlinear relation between the sensor information and the flow field low-dimensional DMD dynamic coefficient is obtained through DNN network training. And finally, reconstructing the full flow field information by the DNN predicted low-dimensional DMD dynamic coefficient and combining with the DMD mode. In the subsequent online prediction stage, the trained model can be used in an actual experiment to realize prediction of unsteady total flow field information from sparse measurement points. The method is simple to implement and is more suitable for engineering application. Compared with the traditional method, the method provided by the invention has better robustness and accuracy. Meanwhile, the invention also has good universality and expandability.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
a method for predicting an all-flow field from sparse sensor information based on a compressed sensing reduction type specifically comprises the following steps:
an off-line phase comprising the steps of:
step S1, obtaining flow field all-state data at different moments through numerical simulation, mapping the data to uniform Cartesian coordinates, and constructing to obtain a flow field data set X;
step S2, selecting partial data from the obtained flow field data set X as data collected by the sensor: si(i ═ 1,2, …, m), where m represents the number of sensors;
step S3, performing DMD decomposition on the obtained flow field data set X to obtain a low-dimensional representation of the flow field data set X, namely a DMD mode phij(j ═ 1,2, …, r) and the corresponding DMD kinetic coefficients bj(j ═ 1,2, …, r), and sparse flooding is usedSelecting DMD mode by dynamic algorithm to obtain dominant DMD mode phi of the flow fieldsp,j(j ═ 1,2, …, n) and the corresponding DMD dynamics factor bsp,j(j ═ 1,2, …, n); wherein r represents the number of acquired DMD modes, and n is the dominant DMD mode number.
Step S4, constructing a full flow field prediction model, wherein the model comprises an LSTM neural network and a deep neural network, and the input of the LSTM neural network is data S which is collected by a sensor at the last moment ti(t) outputting data s collected as a sensor at the predicted time t Δ ti(t + Δ t). The input of the deep neural network is the output of the LSTM neural network, and the output is the DMD dynamic coefficient bsp,j(j ═ 1,2, …, n) predicted values. And (5) training the full flow field prediction model by using the data obtained in the steps S1-S2, so as to obtain the trained full flow field prediction model.
The online prediction stage specifically comprises the following steps:
installing a sensor according to the position of the selected data in the step S2, and collecting sensor data S 'in real time'i(i ═ 1,2, …, m); and inputting the data to a trained full flow field prediction model to obtain a DMD dynamic coefficient b'sp,j(j＝1,2,…,n)；
Predicted DMD kinetic coefficient b'sp,j(j ═ 1,2, …, n) in combination with the DMD mode Φ obtained in the off-line phasesp,j(j ═ 1,2, …, n), to obtain the full state information of the flow field:
X′≈ΦspDαb′sp；
wherein D isαIs a diagonal matrix composed of the amplitude of the DMD as diagonal elements.
Further, in step S1, the flow field data set X includes a velocity field and a pressure field.
Further, the step S2 includes comparing the data SiThe step of normalizing.
Further, in step S2, a flow field wake portion in the obtained flow field data set X is selected as data collected by the sensor.
Further, the step S3 specifically includes the following sub-steps:
step S3.1, decomposing the flow field data set X obtained in step S1 into the following two data sets according to time sequence:
Figure BDA0003143339690000031
wherein: n is the time sequence number of the flow field data set;
step S3.2, singular value decomposition is carried out on the data set X to obtain left and right singular vectors U and V and a characteristic value sigma respectively, and the following formula is shown:
X0＝U∑V*；
wherein denotes a complex conjugate matrix;
step S3.3, calculating to obtain a state matrix of the DMD: fdmd＝U*X1V∑-1；
Step S3.4, for matrix FdmdPerforming eigenvalue decomposition to obtain an eigenvector matrix W of the matrix and a corresponding eigenvalue matrix Lambda;
step S3.5, the DMD mode can be obtained through the eigenvector matrix W, and the calculation formula is as follows:
Φ＝X1V∑-1W
obtaining DMD mode phij(j ═ 1,2, …, r), where r is the number of columns in the matrix of Φ.
S3.6, using the amplitude of the DMD as a diagonal element to construct a diagonal matrix DαAnd obtaining the DMD kinetic coefficient through the characteristic value lambda:
bαVand
wherein, VandA vandermonde matrix that is a characteristic value λ; obtaining the corresponding DMD kinetic coefficient bj(i ═ 1,2, …, r). The amplitude of the DMD is calculated as: α ═ W-1x1。
S3.7, selecting the DMD mode by utilizing a sparse driving DMD algorithm to obtain the dominant DMD mode phi of the flow fieldsp,j(j ═ 1,2, …, n) and the corresponding DMD dynamics factor bsp,j(j＝1,2,…,n)：
Figure BDA0003143339690000032
Wherein the gamma parameter determines the sparsity of the DMD mode,
Figure BDA0003143339690000033
denotes the Frobenius mode, alphajThe amplitude of the jth DMD mode forms a diagonal matrix Dα。
The invention has the beneficial effects that:
1. through a sparsely driven DMD algorithm, the dimension of the flow field can be reduced more effectively in a linear mode, the mode obtained by reducing the order is more beneficial to capturing the time sequence evolution characteristic of the flow field;
2. by combining with the LSTM model, the method provided by the invention further improves the prediction precision and improves the robustness of the method. The invention can accurately predict the all-state information of the flow field even under the condition of high noise.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a sensor layout;
FIG. 3 is a diagram of sensor signals collected during an offline learning phase;
FIG. 4 is a DMD mode diagram selected by the sparsely-driven DMD algorithm;
FIG. 5 is a diagram of sensor signals collected during an online prediction phase;
FIG. 6 is a graph of predicted flow field results provided by the present invention;
FIG. 7 is a comparison graph of the mean square error analysis of the prediction results provided by the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings.
FIG. 1 is a method for realizing pneumatic optimization design by a multi-precision optimization algorithm based on deep learning provided by the invention.
This embodiment is a three-dimensional streaming flow field of a fixed cylinder with reynolds number Re 3900. The wake of the flow field is a fully developed complex turbulent flow field, so that the prediction of all flow information of the complex turbulent flow field from a given sparse measurement point is realized.
The method specifically comprises the following steps:
step S1: in the off-line stage, full-state data of the flow field (velocity field and pressure field) is obtained through CFD direct numerical simulation, and the data is mapped onto uniform cartesian coordinates, in this example, the flow field data is mapped onto 200 × 200 cartesian coordinates, and the interval of the selected flow field time sequence is Δ t ═ 5 × 10-2The training data set adopts 2000 time slices to construct a required flow field data set X;
step S2, in this example, from the obtained flow field data, take x ∈ [1,7 ] in the trail]，y∈[-1,1]Uniformly distributed m data in the range are used as analog data of the sensor
Figure BDA0003143339690000041
Wherein, then, in the prediction, the sensors are averagely arranged in the trail by x epsilon [1,7 ∈]，y∈[-1,1]In the scope, the present embodiment has 32 sensors in total, as shown in fig. 2; wherein, the signals of the 1 st, 6 th, 18 th and 24 th sensors are shown in figure 3;
step S3, performing DMD decomposition on the flow field dataset obtained in step S1, and deriving a low-dimensional representation of the flow field dataset X, i.e., a DMD mode Φj(j ═ 1,2, …, r) and the corresponding DMD kinetic coefficients bj(j is 1,2, …, r), and using sparse driving algorithm to select DMD mode, obtaining dominant DMD mode phi of the flow fieldsp,j(j ═ 1,2, …, n) and the corresponding DMD dynamics factor bsp,j(j＝1,2,…,n)；
The method specifically comprises the following substeps:
step S3.1, decomposing the time sequence flow field data set into the following two data sets:
Figure BDA0003143339690000051
wherein: n is the time sequence number of the flow field data set;
step S3.2, for data set X0Singular Value Decomposition (SVD) is performed to obtain left and right singular vectors U and V, respectively, and a feature value Σ, as shown in the following equation:
X0＝U∑V*；
wherein denotes a complex conjugate matrix;
step S3.3, obtaining a state matrix F of the DMD through a formuladmd＝U*X1V∑-1；
Step S3.4, for matrix FdmdPerforming eigenvalue decomposition to obtain an eigenvector matrix W of the matrix and a corresponding eigenvalue matrix Lambda;
step S3.5, the DMD mode can be obtained through the eigenvector matrix W, and the calculation formula is as follows:
Φ＝X1V∑-1W
obtaining DMD mode phij(j ═ 1,2, …, r), where r is the number of columns in the matrix of Φ.
S3.6, using the amplitude of the DMD mode as a diagonal element to construct a diagonal matrix DαThe amplitude calculation formula of the DMD is as follows: α ═ W-1x1And through the eigenvalue lambda (diagonal element of the eigenvalue matrix lambda), the DMD kinetic coefficient can be obtained:
b＝DαVand
wherein, VandA vandermonde matrix that is a characteristic value λ; obtaining the corresponding DMD kinetic coefficient bj(i＝1,2,…,r)。
Step S3.7, the sparse driving DMD algorithm is to select a dominant DMD mode from all the DMD modes, which can be described as an optimization problem as follows:
Figure BDA0003143339690000052
the gamma parameter determines the sparsity of the DMD mode, i.e., the larger the gamma value is, the fewer the selected DMD modes are. Wherein the content of the first and second substances,
Figure BDA0003143339690000053
denotes the Frobenius mode, alphajThe amplitude of the jth DMD mode forms a diagonal matrix Dα。
All DMD modes and sparsely driven selected DMD modes of the flow field optimized for this example are shown in fig. 4. The number of DMD modes chosen for this example is: 81 flow direction velocity modes, 123 perpendicular to the flow direction velocity modes and 85 pressure field modes;
step S4, constructing a full-flow-field prediction model, where the model includes an LSTM neural network and a deep neural network, where a long-term memory (LSTM) neural network is trained by using information as a sensor signal, and in this embodiment, 3 layers are adopted, and model parameters are as shown in the following table:
Figure BDA0003143339690000061
the LSTM training was then detailed as follows:
data normalization as sensor signals:
Figure BDA0003143339690000062
wherein the content of the first and second substances,
Figure BDA0003143339690000063
which represents the minimum value of the sensor signal,
Figure BDA0003143339690000064
which is indicative of the maximum value of the sensor signal,iis normalized signal;
grouping the data according to a time sequence interval, wherein each group comprises data of a current moment t and a time interval next to the current moment t: { s1(t),s2(t),…sm(t) } and { s }1(t+Δt),s2(t+Δt),…sm(t+Δt)}；
And training the LSTM model by using the grouped data until the output predicted value is converged with the truth loss function. ,the data of the current moment t and the time interval of the current moment t are respectively used as the input and the output of the LSTM model, and the mapping relation between the two data is obtained:
Figure BDA0003143339690000065
then the output value of the LSTM neural network and the DMD dynamic coefficient b obtained in the step S4 are usedsp,j(j-1, 2, …, n) training a Deep Neural Network (DNN) as input and output, respectively, until the predicted value of the output converges with the true loss function. The DNN model parameters used in this example are as follows:
Figure BDA0003143339690000066
in the on-line prediction stage, the full flow field can be predicted by using the trained full flow field prediction model, and the method specifically comprises the following steps:
mounting the sensors in practice, and carrying out mounting arrangement according to the same sensor positions in step S2, as shown in FIG. 2;
acquiring experimental data of the mounted sensor to obtain acquired data s 'of the sensor'i(i ═ 1,2, …, m), in this example, the actual measured data was generated from the raw data at training, superimposed with noise, and the 1 st, 6 th, 18 th, 24 th sensor signals are shown in fig. 5;
predicting the sensor information by using the LSTM model trained in the off-line learning stage;
inputting the predicted sensor signal into a trained DNN network, and predicting a DMD dynamic coefficient b 'from the sensor signal through the DNN network'sp,j(j＝1,2,…,n)；
DMD dynamics coefficient b 'predicted from DNN network'sp,j(j ═ 1,2, …, n) in combination with the DMD mode Φ obtained from the offline phase decompositionsp,j(j ═ 1,2, …, n), to obtain the full state information of the flow field:
X′≈ΦspDαb′sp；
the flow field prediction result obtained in the online learning stage of the embodiment is shown in fig. 6, and it can be seen that even if 24% of experimental error is introduced, a satisfactory result can be obtained for a complex high-Reynolds-number turbulent flow field, and the comparison with a real flow field is very close, which shows that the method provided by the invention can accurately predict the state information of the full flow field from sparse measurement points. Further, fig. 7 shows a quantitative index of prediction accuracy, and a mean square error is used as an evaluation index, and it can be seen from the figure that the influence on the prediction accuracy is very small after different measurement errors are introduced by the method provided by the present invention, which indicates that the method provided by the present invention can robustly predict the state information of the full flow field from the sparse measurement point under different experimental conditions.

Claims (4)

1. A method for predicting an all-flow field from sparse sensor information based on a compressed sensing reduced order model is characterized by specifically comprising the following steps:
an off-line phase comprising the steps of:
step S1, obtaining flow field all-state data at different moments through numerical simulation, mapping the data to uniform Cartesian coordinates, and constructing to obtain a flow field data set
Figure DEST_PATH_IMAGE002
；
Step S2, obtaining flow field data set
Figure 394009DEST_PATH_IMAGE002
And selecting partial data as data collected by the sensor: siI ═ 1,2, …, m, where m represents the number of sensors;
step S3, the obtained flow field data set is used
Figure 357154DEST_PATH_IMAGE002
Performing DMD decomposition to obtain the flow field data set
Figure 798543DEST_PATH_IMAGE002
Low dimensional representation of, i.e. DMD modality
Figure DEST_PATH_IMAGE006
And corresponding DMD kinetic coefficients
Figure DEST_PATH_IMAGE008
And selecting the DMD mode by using a sparse driving algorithm to obtain the dominant DMD mode and the corresponding DMD dynamic coefficient of the flow field
Figure DEST_PATH_IMAGE010
(ii) a Wherein r represents the acquired DMD modal number, and n is the dominant DMD modal number;
the method specifically comprises the following substeps:
step S3.1, decomposing the flow field data set X obtained in step S1 into the following two data sets according to time sequence:
Figure DEST_PATH_IMAGE012
wherein:
Figure DEST_PATH_IMAGE014
the number of time series of flow field data sets;
step S3.2, for the data set
Figure DEST_PATH_IMAGE016
Singular value decomposition is carried out to respectively obtain left and right singular vectors
Figure DEST_PATH_IMAGE018
And
Figure DEST_PATH_IMAGE020
and a characteristic value
Figure DEST_PATH_IMAGE022
As shown in the following formula:
Figure DEST_PATH_IMAGE024
；
wherein denotes a complex conjugate matrix;
step S3.3, calculating to obtain a state matrix of the DMD:
Figure DEST_PATH_IMAGE026
;
step S3.4, to the matrix
Figure DEST_PATH_IMAGE028
Decomposing the eigenvalue to obtain the eigenvector matrix of the matrix
Figure DEST_PATH_IMAGE030
And corresponding eigenvalue matrix
Figure DEST_PATH_IMAGE032
；
Step S3.5, passing the feature vector matrix
Figure 758251DEST_PATH_IMAGE030
The DMD mode can be obtained by the following calculation formula:
Figure DEST_PATH_IMAGE034
obtaining DMD modality
Figure DEST_PATH_IMAGE035
Wherein r is
Figure DEST_PATH_IMAGE037
The number of columns of the matrix;
s3.6, using the amplitude of the DMD as a diagonal element to construct a diagonal matrix
Figure DEST_PATH_IMAGE039
And passing the characteristic value
Figure DEST_PATH_IMAGE041
Obtaining the DMD kinetic coefficient:
Figure DEST_PATH_IMAGE043
wherein is a characteristic value
Figure 875419DEST_PATH_IMAGE041
The vandermonde matrix of (a); obtaining corresponding DMD kinetic coefficients
Figure DEST_PATH_IMAGE044
The amplitude calculation formula of the DMD is as follows:
Figure DEST_PATH_IMAGE046
；
s3.7, selecting the DMD mode by utilizing a sparse driving DMD algorithm to obtain the dominant DMD mode of the flow field
Figure DEST_PATH_IMAGE048
And corresponding DMD kinetic coefficients
Figure DEST_PATH_IMAGE049
：
Figure DEST_PATH_IMAGE051
Wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE053
the parameters determine the sparseness of the DMD modes,
Figure DEST_PATH_IMAGE055
represents a Frobenius model, and represents a Frobenius model,
Figure DEST_PATH_IMAGE057
for the amplitude of the jth DMD mode, form a diagonal matrix
Figure 91024DEST_PATH_IMAGE039
；
Step S4, constructing a full flow field prediction model, wherein the model comprises an LSTM neural network and a deep neural network, and the input of the LSTM neural network is data collected by a sensor at the last moment t
Figure DEST_PATH_IMAGE059
The output is the predicted time
Figure DEST_PATH_IMAGE061
As data collected by the sensor
Figure DEST_PATH_IMAGE063
(ii) a The input of the deep neural network is the output of the LSTM neural network, and the output is DMD dynamic coefficient
Figure 405329DEST_PATH_IMAGE049
Predicting a value; training the full flow field prediction model by using the data obtained in the steps S1-S2 so as to obtain a trained full flow field prediction model;
the online prediction stage specifically comprises the following steps:
installing the sensor according to the position of the selected data in the step S2, and collecting the sensor data in real time
Figure DEST_PATH_IMAGE065
(ii) a And inputting the data to a trained full flow field prediction model to obtain DMD dynamic coefficient
Figure DEST_PATH_IMAGE067
；
DMD kinetic coefficients to be predicted
Figure 227398DEST_PATH_IMAGE067
DMD modality incorporating offline phase acquisition
Figure 771643DEST_PATH_IMAGE048
And therefore, acquiring all-state information of the flow field:
Figure DEST_PATH_IMAGE069
；
wherein the content of the first and second substances,
Figure 748695DEST_PATH_IMAGE039
is a diagonal matrix composed of the amplitude of the DMD mode as diagonal elements.
2. The method of claim 1, wherein in step S1, the flow field data set
Figure 882786DEST_PATH_IMAGE002
Including velocity fields, pressure fields.
3. The method according to claim 1, wherein the step S2 further comprises comparing the data
Figure DEST_PATH_IMAGE071
The step of normalizing.
4. The method of claim 1, wherein in step S2, the flow field data set is obtained from
Figure 603486DEST_PATH_IMAGE002
The wake part of the intermediate flow field is selected as the data collected by the sensor.
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