CN113268822A - Centrifugal pump performance prediction method based on small sample nuclear machine learning - Google Patents

Abstract

A centrifugal pump performance prediction method based on small sample nuclear machine learning belongs to the technical field of centrifugal pump performance prediction and mainly comprises the following steps: (1) performing feature selection and standardization processing on the collected sample data; (2) constructing a small sample kernel machine learning Gaussian process regression prediction model; (3) selecting a suitable nonlinear kernel function; (4) training unknown hyper-parameters of the model based on the training data; (5) the validity of the model is verified based on the test data. The method takes the impeller structure and design parameters as basic data, learns the nonlinear relation between the impeller parameters and the lift and the efficiency of the centrifugal pump through a kernel machine learning Gaussian process regression model, and further realizes the prediction of the performance of the centrifugal pump. The invention fully and secondarily utilizes the design data of the existing centrifugal pump, has low requirements on the number of training samples by the prediction model, has low model construction difficulty, high model precision and good stability, and is more suitable for rapidly predicting the performance of the centrifugal pump in engineering design and optimization.

Description

A centrifugal pump performance prediction method based on small sample kernel machine learning

technical field

The invention belongs to the field of centrifugal pump performance prediction, and specifically relates to a centrifugal pump performance prediction method based on small sample kernel machine learning.

Background technique

As a general-purpose machine, centrifugal pumps are widely used in many industrial fields. The impeller is one of the most important hydraulic components in the centrifugal pump. It converts the mechanical energy of the motor into the energy of the liquid, and then realizes the transportation of the liquid. The rational design of the impeller is very important to the performance of the entire pump. The impeller has many structural parameters such as outlet width, number of blades, outer diameter of the impeller, etc. The change of some of these parameters will lead to many complex flow phenomena inside the impeller during the rotation of the impeller, such as secondary flow, wake jet, eddy current, etc. , some of these flow structures consume the energy of the liquid inside, which in turn affects the performance of the centrifugal pump, such as head and efficiency. Due to the complexity of the internal flow of the impeller, there is often a complex nonlinear relationship between the impeller structural parameters, internal flow and external characteristics. It becomes difficult to design or optimize the design, and this is when preliminary prediction of centrifugal pump performance becomes very important.

At present, the common performance prediction methods of centrifugal pumps mainly include flow field analysis method based on numerical simulation, empirical statistical method and data-driven machine learning method. In the modern design of the pump, the performance of the pump is generally estimated based on the numerical simulation method, but for the physical model with more complex structure, the calculation cost is unbearable and the process is cumbersome. The empirical statistical method calculates various losses in the pump through semi-empirical and semi-theoretical formulas, and finds the relationship between various losses and pump structural parameters after certain simplification and assumption of convection action, and then establishes a hydraulic loss prediction model. Due to the differences between various pump types, this method is not universal and generalizable.

Gaussian Process Regression (GPR) is a data-based kernel machine learning method, which is not only suitable for nonlinear prediction under small samples, but also has many other advantages, such as it not only provides predicted values but also provides The variance is used to evaluate the uncertainty of the prediction; the hyperparameters are not very sensitive to the prediction results; as a kernel machine learning method, there are many kernel functions that can adapt to data with different characteristics, and the corresponding kernel functions can be developed according to the data characteristics. Due to the high construction efficiency of GPR, few training samples required, and strong nonlinear relationship learning ability, its application in centrifugal pump performance prediction can greatly improve the design efficiency of centrifugal pump and reduce its performance prediction time.

SUMMARY OF THE INVENTION

According to the above, the purpose of the present invention is to provide a centrifugal pump performance prediction method based on small sample kernel machine learning, which can more efficiently and accurately predict the hydraulic performance of the centrifugal pump according to the impeller parameters with the support of a small number of samples.

The present invention provides the following technical solutions: a method for predicting the performance of a centrifugal pump based on small sample kernel machine learning, comprising the following steps.

(1) Standardize the collected sample data.

(2) Construct a small sample kernel machine learning Gaussian process regression prediction model.

(3) Choose a suitable nonlinear kernel function.

(4) The unknown hyperparameters of the model are trained based on the training data.

(5) Check the validity of the model based on the test data.

In the above scheme, in the step (1), in order to reduce the training time of the prediction model and ensure its prediction accuracy, the collected sample data is standardized, so that the sample data satisfies the standard normality with a mean value of 0 and a variance of 1. distribution, the specific formula is as follows:

为标准化后的数据，x_i为原始数据，μ为同一维度数据的均值，σ为同一维度数据的方差。in,

is the standardized data, x _i is the original data, μ is the mean of the data in the same dimension, and σ is the variance of the data in the same dimension.

In the above scheme, in the step (2), according to the mathematical principle of Gaussian process regression, the prior distribution between the known training samples and the unknown test samples is constructed in MATLAB, and the formula is as follows:

表示噪声，I表示单位矩阵。Where y represents the set of training sample output variables; f _* represents the output of the unknown test sample; K(X, X) represents the kernel function relationship between the input variables of the training sample, K(X, x _* ) and K(x _* , X) all represent the kernel function relationship between the training sample input variables and the test sample input variables, K(x _* ,x _* ) represents the kernel function relationship between the test sample input variables,

represents noise, and I represents the identity matrix.

In the above scheme, the posterior distribution of the unknown test sample output variable can be expressed as

为该后验分布的均值，其数值可代表测试样本的未知输出变量，cov(f_*)为该后验分布的方差，可表征该输出变量的不确定性，它们可分别表示为以下两式：in

is the mean of the posterior distribution, and its value can represent the unknown output variable of the test sample, cov(f _* ) is the variance of the posterior distribution, which can represent the uncertainty of the output variable, and they can be expressed as the following two formulas respectively :

In the above scheme, in the step (3), by comparing three commonly used Squared Exponential (SE) nonlinear kernel functions, rational quadratic (Rational Quadratic, RQ) nonlinear kernel functions and Matern5/2 nonlinear kernel functions. The performance of the kernel function, and finally the SE kernel function is used to construct the nonlinear relationship between the impeller parameters and the performance of the centrifugal pump. The formula is as follows:

称为信号方差，控制核函数的输出大小；

l称为特征长度，控制输入变量各维度的特征属性对输出结果的影响程度。in

Called the signal variance, it controls the output size of the kernel function;

l is called the feature length, which controls the degree of influence of the feature attributes of each dimension of the input variable on the output result.

In the above scheme, in the step (4), the initial value of the unknown hyperparameter is randomly given, and the optimal value of the hyperparameter can be obtained through training. The objective function with respect to the unknown hyperparameters in training Gaussian process regression can be expressed as

包含了模型中的未知参数，通过梯度下降法求解目标函数NLML关于每一个未知超参数偏导数的最小值即可得到最优的超参数值。in

The unknown parameters in the model are included, and the optimal hyperparameter value can be obtained by solving the minimum value of the partial derivative of the objective function NLML with respect to each unknown hyperparameter by the gradient descent method.

In the above scheme, in the step (5), after the optimal value of the hyperparameter is obtained by training, the impeller parameter value of the standardized test sample is input into the model, and the corresponding centrifugal pump performance can be obtained after the model completed by training. The mean and variance of the values are de-standardized. The de-standardized mean can represent the prediction result of centrifugal pump performance, and the de-standardized variance can represent the uncertainty of the prediction result. The accuracy and validity of the proposed centrifugal pump prediction model can be tested by calculating the relative error between the predicted performance value of the centrifugal pump and the real experimental value.

Beneficial effects of the present invention: the performance prediction method of centrifugal pump based on small sample kernel machine learning realizes the nonlinear learning between centrifugal pump impeller parameters and centrifugal pump performance flexibly according to different kernel functions in Gaussian process under the support of a small number of training samples relationship, and then accurately predict the performance of the centrifugal pump. The proposed prediction model needs to set fewer unknown parameters, and the value of the unknown parameters has little influence on the prediction results, the construction efficiency of the model is greatly improved, and it is more suitable for the performance prediction of centrifugal pumps in engineering practice.

Description of drawings

FIG. 1 is a flow chart of a method for predicting the performance of a centrifugal pump according to the present invention.

Figure 2 is a comparison chart of the predictive ability of three different nonlinear kernel functions.

Figure 3 is a comparison chart of the predicted lift results of the 14 groups of test samples and the experimental values.

Figure 4 is a comparison diagram of the efficiency prediction results of 14 groups of test samples and the experimental values.

Figure 5 is a comparison chart of the lift prediction results obtained by 14 groups of test samples through four different methods.

Figure 6 is a comparison chart of the efficiency prediction results obtained by four different methods for 14 groups of test samples.

Detailed ways

In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments of the description. It should be understood that the specific embodiments described herein are only used to explain the present invention, but not to limit the present invention.

On the contrary, the present invention covers any alternatives, modifications, equivalents and arrangements within the spirit and scope of the present invention as defined by the appended claims. Further, in order to give the public a better understanding of the present invention, some specific details are described in detail in the following detailed description of the present invention. The present invention can be fully understood by those skilled in the art without the description of these detailed parts.

(1) Standardize the collected sample data. 68 groups of impeller hydraulic model data with different specific speeds are collected through Modern Pump Theory and Design as the entire sample set. The model input variables can be specific speed, impeller speed, impeller outlet width, number of blades and other impeller parameters, and the output variables can be Performance parameters of centrifugal pump head, efficiency, etc. From the entire sample set, 14 sets of sample data from small to large specific speeds are selected as test samples, and the rest of the samples are used as training samples. Standardize the entire sample set so that the same attribute of all samples satisfies a normal distribution with a mean of 0 and a variance of 1. Some sample data are shown in Table 1.

Table 1 Partial centrifugal pump sample data

(2) Build a prediction model. According to the mathematical principle of Gaussian process regression of small sample kernel machine learning, the posterior distribution is obtained from the prior distribution, and the corresponding program is compiled in MATLAB to obtain its prediction model. Three kernel functions with strong nonlinear learning ability in Gaussian process are selected to learn the nonlinear relationship between centrifugal pump impeller parameters and centrifugal pump head and efficiency. The three kernel functions are SE kernel function, RQ kernel function and Matern5/2 kernel function. The performance prediction results of centrifugal pump based on three kernel functions were compared and analyzed, and the most suitable kernel function for learning the relationship between centrifugal pump impeller parameters and centrifugal pump performance was selected. Figure 2 shows four statistical indicators that reflect the learning ability and prediction accuracy of the kernel function, namely the root mean square error (RMSE), the mean relative error (MAPE), the maximum relative error (MARE) and the Coefficient of Determination (R ² ). It can be seen from Figure 2 that the Gaussian process regression prediction model based on the SE kernel function has the highest prediction accuracy for the head and efficiency of the centrifugal pump. Therefore, the SE kernel function is selected as the kernel function for learning the relationship between the impeller parameters of the centrifugal pump and the performance of the centrifugal pump. .

(3) Prediction model training. First, the initial value of the unknown hyperparameter is given. Due to the good adaptability of the Gaussian process regression model, the initial value of the unknown hyperparameter will not have a sensitive impact on the performance prediction result of the centrifugal pump, so when the mean value is 0, 20 groups of initial values of unknown hyperparameters are randomly assigned in the space with a variance of 2 normal distribution, and then based on the training data, the unknown hyperparameters are optimized and trained by the gradient descent algorithm. The objective function of this optimization is

包含了模型中的未知超参数。待20组超参数初值均寻优完成后，选择使目标函数NLML达到最小的那组未知超参数初值作为该预测模型进行离心泵性能预测的初值。in

Contains unknown hyperparameters in the model. After the optimization of the initial values of the 20 groups of hyperparameters is completed, the initial value of the unknown hyperparameters that minimizes the objective function NLML is selected as the initial value for the prediction model to predict the performance of the centrifugal pump.

(4) Predictive model validity test. After the training of the prediction model is completed, the nonlinear relationship between the impeller parameters of the centrifugal pump and the head and efficiency of the centrifugal pump in the training samples provided has been basically learned, and the impeller parameters of the 14 groups of test samples determined in the first step are used as predictions The input data of the model can be used to predict the mean and variance of the head and efficiency corresponding to the 14 groups of impeller parameters, and de-standardize the 14 groups of mean and variance of the initial output. The de-standardized mean can represent the difference between the head and efficiency. Predicted value, the variance can represent the uncertainty of the obtained prediction result. Figures 3 and 4 show the comparison between the predicted head and efficiency and their experimental values, respectively. The shaded area in the figure represents the 95% confidence interval transformed from the predicted variance. The narrower the range, the uncertainty of the predicted results. The smaller, the higher the reliability.

As can be seen from Fig. 3, the predicted value of head agrees well with its experimental value, and its 95% confidence interval is narrow enough in the first 10 test samples, while in the last four test samples, due to the increase of its specific speed, There is relatively little training data in the large specific speed range, which in turn leads to large uncertainty in prediction. From the prediction of the efficiency in Figure 4, compared with the prediction result of the head, the prediction accuracy of the efficiency is slightly lower, the prediction value of the efficiency is basically lower than the experimental value, and the change law of the prediction value and the experimental value is basically the same of. The uncertainty of efficiency prediction is relatively large, but the experimental value is basically within the 95% confidence interval obtained by the efficiency prediction, indicating that the confidence interval includes its true value, so the prediction result is still credible. The absolute relative errors between the predicted and experimental values of head and efficiency for the 14 test samples were further calculated. Among all the tested samples, the maximum absolute relative error between the predicted value of head and the experimental value is 6.66%, and most of the absolute relative errors are less than 4%, and the maximum absolute relative error between the predicted value of efficiency and the experimental value is 10.54% , and most of the absolute relative errors are less than 8%, and the prediction accuracy meets the prediction requirements of centrifugal pumps in practical engineering design.

Table 2 shows the performance prediction method of centrifugal pump based on small sample kernel machine learning proposed by the present invention and three common prediction models BP neural network (Back Propagation Neural Network, BPNN), Radial Basis Function Neural Network (Radial Basis Function Neural Network) Network) and support vector regression (Support VectorRegression, SVR). From the comprehensive results of the four performance evaluation indicators shown in Table 2, under the same training samples, GPR has the highest accuracy and the best stability in the prediction of centrifugal pump head and efficiency among the four models.

Table 2 Performance comparison of four centrifugal pump prediction models

Figures 5 and 6 visually show the comparison between the predicted and experimental values of head and efficiency for the four models. From the head comparison, in the test data, the predicted values of the four models are basically consistent with the experimental values, indicating that the nonlinear relationship between the input variables and the head is fully learned, and the defined input variables can be very A good representation of the influence characteristics of head. From the perspective of efficiency comparison, the change law of the model predicted value and the experimental value is basically the same. On the whole, the head and efficiency predicted by GPR are closer to their experimental values, indicating that the prediction accuracy is higher.

Compared with the common performance prediction method of centrifugal pump based on numerical simulation, the method of using impeller parameters to predict the performance of centrifugal pump based on the GPR model can make secondary use of the existing data in engineering design application to realize the prediction of centrifugal pump performance. The cycle is short, the difficulty is low, and the universality is strong. Compared with other common data-based prediction models, in centrifugal pump performance prediction, the GPR-based prediction model requires less training data, which reduces the difficulty of data collection; The influence of its prediction performance is small, so the training is easier; the stability of the GPR model is good, it can not only provide the prediction result, but also provide the degree of uncertainty corresponding to the prediction result, and its reliability is stronger.

To sum up, the above-mentioned embodiments are not limiting embodiments of the present invention, and any modifications or equivalent deformations made by those skilled in the art on the basis of the essential content of the present invention are all within the technical scope of the present invention.

Claims (7)

1. A centrifugal pump performance prediction method based on small sample nuclear machine learning is characterized by comprising the following steps:

(1) performing feature selection and standardization processing on the collected sample data;

(2) constructing a small sample kernel machine learning Gaussian process regression prediction model;

(3) selecting a suitable nonlinear kernel function;

(4) training unknown hyper-parameters of the model based on the training data;

(5) the validity of the model is verified based on the test data.

2. The method according to claim 1, wherein in step (1), in order to reduce the training time of the prediction model and ensure the prediction accuracy thereof, the collected sample data is normalized so that the sample data satisfies a standard normal distribution with a mean value of 0 and a variance of 1, and the specific formula is as follows:

wherein,

for normalized data, x_iFor raw data, μ is the mean of data of the same dimension, and σ is the variance of data of the same dimension.

3. The method for predicting the performance of the centrifugal pump based on the small sample nuclear machine learning as claimed in claim 1, wherein in the step (2), the prior distribution between the known training sample and the unknown test sample is constructed in MATLAB software according to the mathematical principle of gaussian process regression, and the formula is as follows:

wherein y represents a set of training sample output variables; f. of_*An output representing an unknown test sample; k (X, X) denotes the kernel functional relationship between the training sample input variables, K (X, X)_*) And K (x)_*X) each represent a kernel function relationship between a training sample input variable and a test sample input variable, K (X)_*,x_*) Representing the kernel function relationship between the test sample input variables,

representing noise and I representing an identity matrix.

4. The method of claim 1, wherein the posterior distribution of the unknown test sample output variables is expressed as

Wherein

The mean of the posterior distribution whose value represents the unknown output variable of the test sample, cov (f)_*) For the variance of the posterior distribution, the uncertainty of the output variable can be characterized, which can be expressed as the following two equations, respectively:

5. the method for predicting centrifugal pump performance based on small sample nuclear machine learning of claim 1, wherein in the step (3), the SE kernel is finally used to construct the nonlinear relationship between the impeller parameters and the centrifugal pump performance by comparing the performances of three commonly used Square Exponential (SE) nonlinear kernels, Rational Quadratic (RQ) nonlinear kernels and Matern5/2 nonlinear kernels, and the formula is as follows:

wherein

Called signal variance, controlling the output magnitude of the kernel function;

l is called the characteristic length and controls the influence degree of the characteristic attribute of each dimension of the input variable on the output result.

6. The method for predicting the performance of the centrifugal pump based on the small sample nuclear machine learning as claimed in claim 1, wherein in the step (4), the initial value of the unknown hyper-parameter is randomly given, and the optimal value of the hyper-parameter can be obtained through training. An objective function for an unknown hyperparameter in a training Gaussian process regression can be expressed as

Wherein

The unknown parameters in the model are included, and the minimum value of the objective function NLML relative to each unknown hyper-parameter partial derivative is solved through a gradient descent method, so that the optimal hyper-parameter value can be obtained.

7. The method for predicting the performance of the centrifugal pump based on the small-sample nuclear machine learning as claimed in claim 1, wherein in the step (5), after the optimal value of the hyper-parameter is obtained through training, the impeller parameter value of the standardized test sample is input into the model, the mean value and the variance of the corresponding performance value of the centrifugal pump can be obtained through the trained model, the mean value and the variance are subjected to anti-standardization, the mean value after the anti-standardization can represent the prediction result of the performance of the centrifugal pump, and the variance after the anti-standardization can represent the uncertainty of the prediction result; the accuracy and the effectiveness of the proposed centrifugal pump prediction model can be checked by calculating the relative error between the predicted centrifugal pump performance value and the real experimental value.

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