CN117952021A - Sequence dotting optimization method based on proxy model - Google Patents

Abstract

The invention discloses a sequence dotting optimization method based on a proxy model, which belongs to the field of proxy model optimization and comprises the following steps of S1, sampling in an input space to obtain a sample set; s2, training the agent model, and selecting random numbers; s3, judging whether the random number is larger than a scale factor, if so, sampling in an input space by adopting an MSP (MSP) criterion, otherwise, sampling in the input space by adopting a Maximin criterion; s4, adding the sampled sample points into a sample set, judging whether the sample set meets a termination condition, outputting a sample and a maximum value or a minimum value if the sample set meets the termination condition, and otherwise, entering S5; s5, detecting whether the sample set has clusters, if so, entering S6, otherwise, returning to S2; s6, adding the hyperspheres formed by the clustered samples as constraint conditions into the two criteria in the S3, and returning to the S2. The sequence dotting optimization method of the scheme has the advantages that a larger initial sample size is not needed, a dotting criterion does not depend on a specific agent model construction method, local optima are not easy to fall in, and the like.

Description

Sequence dotting optimization method based on proxy model

Technical Field

The invention belongs to the field of agent model optimization methods, and particularly relates to a sequence dotting optimization method based on an agent model.

Background

Numerical simulation models are often relied upon in engineering designs to predict the performance of the system in order to explore the design space and find optimal designs, such as optimization to simplify the reaction kinetics mechanism. The genetic algorithm is easy to realize the optimization of simplifying the reaction dynamics mechanism, and is convenient to tune, so that the genetic algorithm becomes the most commonly used solution to the specific optimization problem. However, genetic algorithm is used as a random global search optimization, the sample size required by the search process is huge, and a large amount of numerical simulation calculation is required to be called for simplifying the optimization problem of the reaction dynamics mechanism, so that the calculation cost is increased sharply. Agent model based optimization has proven to be an effective method of searching for optimal designs. The proxy model replaces the costly simulation model by approximating the input-output response and finds the optimal design based on this proxy model, a process called proxy model-based optimization.

The optimization problem of simplifying the reaction kinetics mechanism also adopts the method as a means for searching the optimal design. However, the conventional agent model-based optimization method builds an agent model from these sample points and searches for an optimal design by sampling the input space at one time. Such methods typically require a large number of samples, otherwise the proxy model has insufficient accuracy and cannot be searched for an optimal design; even if the overall accuracy of the proxy model meets the requirement, it is difficult to ensure that the model accuracy at the optimal design position meets the same requirement, so that it is difficult to search for a true optimal design.

The existing optimization method based on the agent model has the improvement scheme of sequential dotting, namely small-scale sampling is firstly carried out in the whole input space, and the sampling is carried out in the input space in a targeted manner according to the characteristics of each round of agent model. Pertinence here refers to iterative dotting according to certain criteria, such as improving the desired criteria, improving the probability criteria, etc., which are commonly used, but which rely on the Bayesian inference-based proxy model construction method, which fail once other modeling methods are used.

Disclosure of Invention

Aiming at the defects in the prior art, the sequence dotting optimization method based on the proxy model solves the problem that the existing sequence dotting criterion needs to be built by depending on a specific proxy model.

In order to achieve the above summary, the present invention adopts the following technical scheme:

The sequence dotting optimization method based on the proxy model comprises the following steps:

S1, acquiring an optimization problem of an object to be optimized, determining the dimension and the boundary of an input space based on the optimization problem, and sampling the input space by using a Latin hypercube method to obtain a sample set;

S2, training a proxy model with regression capability by adopting a sample set, calculating a scale factor by adopting an arctangent function, and then selecting a random number between 0 and 1;

S3, judging whether the random number is larger than a scale factor, if so, sampling in an input space by adopting a maximum/small agent model prediction criterion, otherwise, sampling in the input space by adopting a maximum minimum distance criterion;

S4, adding the sampled sample points into a sample set, judging whether the sample set meets a termination condition, if so, outputting a maximum value or a minimum value obtained by the sample set and the agent model, otherwise, entering a step S5;

S5, detecting whether a clustering phenomenon exists in the sample set, if so, entering a step S6, otherwise, directly returning to the step S2;

S6, taking the hypersphere formed by the clustered samples as a current reject domain, adding the hypersphere as a constraint condition into a maximum/small agency model prediction criterion and a minimum distance criterion based on maximization, and returning to the step S2.

Further, the method for detecting whether the clustering phenomenon exists in the sample set comprises the following steps:

statistical sample set Any one of samples/>Number M of other samples present nearby:

M=

Wherein, To calculate the number of elements in the collection; /(I)For any sample in the sample set,/>For non-sample concentrationAny sample outside; /(I)Is a 2-norm; /(I)Is the dimension of the input space; /(I)Is the maximum value;

Judgment sample Whether the number M of other samples existing nearby is greater than or equal to a clustering threshold, if so, the samples/>Clustering phenomenon exists nearby; otherwise sample/>No clustering phenomenon exists nearby.

The technical scheme has the beneficial effects that in the subsequent sampling process, the searched area is avoided, and the method is prevented from being repeatedly sampled in the same area to be trapped in local optimum.

Further, the hypersphere formed by the clustered samples is formed by the samplesIs the sphere center,/>Is a sphere formed by radius;

the constraint conditions are as follows: Wherein/> Is a sample in the sample set.

Further, the termination condition is that a standard deviation of a predicted optimal solution of the proxy model in a plurality of sequence addition points is less than a standard deviation threshold, and a total sample size is greater than a sample size threshold:

Wherein, Standard deviation for predicting optimal solution; /(I)The optimal solution is predicted for the point adding of the agent model in the current last k round; /(I)The average value of the optimal solution of the prediction of the agent model from the current nth round to the last k round; setting a standard deviation threshold value; /(I) Is the total sample size; /(I)The method comprises the steps of sampling an initial sample size in an input space by using a Latin hypercube method; /(I)For adding number of points,/>Is a sample size threshold; k is a round variable; n is the total number of proxy model updates.

The technical scheme has the beneficial effects that whether to terminate the method can be determined according to the convergence condition of the method in the sequence dotting process and the total calculation amount acceptable by the actual engineering problem.

Further, the Latin hypercube method is adopted to sample the sample size in the input spaceThe method comprises the following steps:

Wherein, For input space,/>And/>Input space/>, respectivelyLower and upper bounds of (2); n is/>Is a dimension of (2); and/> Sample/>, respectivelyLower and upper bounds of (2); /(I)Is a proportional symbol; /(I)Is the dimension of the input space.

Further, the maximum/small agent model prediction criterion is searching the maximum/small value of the agent model prediction value in the whole input space, and the optimization problem is that:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) Is an input space.

Further, maximizing the minimum distance criterion is to increase the input space sample filling rate and to set the samplesTaking a boundary formed by samples meeting the conditions as a constraint condition, and sampling; the samples satisfying the conditions are:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) And/>True values/>, respectively, corresponding to samples xMaximum and minimum of (a)/>Is a scale factor; /(I)Samples that meet the condition; is a sample set;

The optimization problem of maximizing the minimum distance criterion is:

Wherein, Any sample in the sample set; /(I)And/>Respectively/>Lower bound and upper bound of (2).

The technical scheme has the beneficial effects that the maximum/small agent model prediction criterion realizes local development by optimizing on the agent model, and the maximum minimum distance criterion realizes global exploration by increasing the filling rate of the input space sample. Both sequence dotting criteria are easy to implement and do not depend on the type of proxy model.

Further, the expression of the arctangent function is:

Wherein, Is a scale factor; /(I)Is the sample size threshold.

Compared with the point adding optimization method in the prior art, the method has the beneficial effects that:

1) Independent of the type of the agent model, various agent models can be optimized by using the scheme; two sequence dotting criteria are used: the maximum/small agent model prediction criterion is to apply an optimization algorithm to search and predict maximum/small values on the agent model, and only the agent model has the function of receiving input parameters and predicting output parameters, and is independent of the type of the agent model; the maximization minimum distance criterion is based on the input space sample filling rate, and is only related to the distribution of the samples in the input space, and is independent of the type of the proxy model.

2) In the process of adding the point optimization, the clustering phenomenon of each sample point is detected, so that a reject domain is constructed, and the reject domain is added into the search criterion of the scheme, so that the reject domain can be avoided during each search, the search domain is separated from a local optimal area in time, and the situation of sinking into the local optimal during the point adding search is avoided.

3) When the clustering phenomenon occurs, the corresponding reject domain is used as a constraint condition, the search domain is reduced by the constraint condition which is continuously increased, the sampling can be performed in the optimal design area as much as possible, and the real optimal design can be effectively searched by the smaller total sample size.

Drawings

FIG. 1 is a flow chart of a proxy model-based sequence dotting optimization method.

Fig. 2 is a graph showing the relationship between the optimal value and the number of samples at the present number of samples.

Detailed Description

The following description of the embodiments of the present invention is provided to facilitate understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and all the inventions which make use of the inventive concept are protected by the spirit and scope of the present invention as defined and defined in the appended claims to those skilled in the art.

Referring to FIG. 1, FIG. 1 shows a flow chart of a proxy model-based sequential point-wise optimization method; as shown in FIG. 1, it includes steps S1-S6.

In step S1, an optimization problem of an object to be optimized is obtained, the dimension and the boundary of an input space are determined based on the optimization problem, and a Latin hypercube method is adopted to sample the input space, so that a sample set is obtained;

The object to be optimized in the scheme can be the mechanism optimization of the fuel in the reaction dynamics, and when the object to be optimized is the total package reaction dynamics mechanism of kerosene (specific description can refer to paper Simulation of kerosene fueled RBCC Engine based on SKELETAL MECHANISM, chinese translation of the text: simulation of kerosene RBCC engine based on skeleton mechanism, author: liu Bing, conference: 21st AIAA International Space Planes and Hypersonics Technologies Conference, publication date: 2017-03), the optimization problem is: in thermodynamic state space is temperature T epsilon [1000, 2000] K, pressure P epsilon [0.5, 10] atm, equivalence ratio In the range of E [0.5, 2.0], taking the dynamic parameter change multiple of the total package mechanism as an input space, uniformly taking 5X 4 sample points in a defined thermodynamic state space, and calculating the average value of the ignition delay time relative errors between the total package mechanism and the skeleton mechanism (the specific description can refer to paper RP-3 aviation kerosene alternative fuel and chemical reaction dynamic model thereof, author: zheng Dong and the like, journal: physical chemistry journal, publication date: 2015-01) as an optimization objective function, wherein the optimization purpose is to obtain the minimum value of the objective function.

In one embodiment of the invention, latin hypercube method is used to sample the sample size in the input spaceThe method comprises the following steps:

Wherein, For input space,/>And/>Input space/>, respectivelyLower and upper bounds of (2); n is/>Is a dimension of (2); and/> Sample/>, respectivelyLower and upper bounds of (2); /(I)The proportional symbol is a proportional proportion, and the proportional proportion is generally set to be 2-10; /(I)Is the dimension of the input space.

The scheme uses the sample sizeThe initial scale of the Latin hypercube sampling can be controlled, and when the input space dimension is higher, the initial scale is larger, so that the proper initial sampling space filling rate is ensured to be generated. Because the method adopts a sequence dotting strategy, the initial sample number does not need to be excessively large, and therefore, the scale of the initial sample only needs to be proportional to the dimension of the input space and does not need to be exponentially increased along with the dimension.

In the step S2, training a proxy model with regression capability by adopting a sample set, calculating a scale factor by adopting an arctangent function, and then selecting a random number between 0 and 1; the expression of the arctangent function is:

Wherein, Is a scale factor; /(I)Is the sample size threshold.

The agent model with regression capability can be a polynomial regression model, a support vector machine or a Gaussian process regression model.

In step S3, judging whether the random number is larger than a scale factor, if so, sampling in an input space by adopting a maximum/small agent model prediction criterion, otherwise, sampling in the input space by adopting a maximum minimum distance criterion;

The maximum/small agent model prediction criterion is to search the maximum/small value of the agent model prediction value in the whole input space, and the optimization problem is as follows:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) Is an input space.

Further, maximizing the minimum distance criterion is to increase the input space sample filling rate and to set the samplesTaking a boundary formed by samples meeting the conditions as a constraint condition, and sampling; the samples satisfying the conditions are:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) And/>True values/>, respectively, corresponding to samples xMaximum and minimum of (a)/>Is a scale factor; /(I)Samples that meet the condition; is a sample set;

The optimization problem of maximizing the minimum distance criterion is:

Wherein, Any sample in the sample set; /(I)And/>Respectively/>Lower bound and upper bound of (2).

In step S4, adding the sampled sample points into a sample set, judging whether the sample set meets a termination condition, if so, outputting a maximum value or a minimum value obtained by the sample set and the agent model, otherwise, entering step S5;

when the scheme is implemented, the preferable termination condition is that the standard deviation of the predicted optimal solution of the agent model in a plurality of sequence adding points is smaller than a standard deviation threshold value, and the total sample size is larger than a sample size threshold value:

Wherein, Standard deviation for predicting optimal solution; /(I)The optimal solution is predicted for the point adding of the agent model in the current last k round; /(I)The average value of the optimal solution of the prediction of the agent model from the current nth round to the last k round; setting a standard deviation threshold value; /(I) Is the total sample size; /(I)The method comprises the steps of sampling an initial sample size in an input space by using a Latin hypercube method; /(I)For adding number of points,/>Is a sample size threshold; k is a round variable; n is the total number of proxy model updates.

In step S5, detecting whether a clustering phenomenon exists in the sample set, if so, entering step S6, otherwise, directly returning to step S2;

In implementation, the method for detecting whether the clustering phenomenon exists in the sample set preferably comprises the following steps:

statistical sample set Any one of samples/>Number M of other samples present nearby:

M=

Wherein, To calculate the number of elements in the collection; /(I)For any sample in the sample set,/>For non-sample concentrationAny sample outside; /(I)Is a 2-norm; /(I)Is the dimension of the input space; /(I)Is the maximum value;

Judgment sample Whether the number M of other samples existing nearby is greater than or equal to a clustering threshold, if so, the samples/>Clustering phenomenon exists nearby; otherwise sample/>No clustering phenomenon exists nearby.

In step S6, the hypersphere formed by the clustered samples is taken as the current reject domain, and is added as a constraint condition to the maximum/small agent model prediction criterion and the minimum distance criterion based on the maximization, and then step S2 is returned.

Wherein, the hypersphere formed by the clustered samples is formed by the samplesIs the sphere center,/>Is a sphere formed by radius;

the constraint conditions are as follows: Wherein/> Is a sample in the sample set.

The following describes a sequential dotting optimization method according to the scheme by taking optimization of total package mechanism of fuel in reaction dynamics as an optimization object:

the general package reaction kinetics model of kerosene (hereinafter referred to as general package mechanism) can be specifically referred to in table 1:

Table 1 table of the total package mechanism kinetics parameters for kerosene

Parameters A, b and E listed in Table 1 are mainly used in the Arrhenius formulaTo calculate the forward rate constant of the reaction, e is the natural logarithm, T is the temperature, and R is the ideal gas constant (8.314J/(mol. K)) in the Arrhenius formula. The ignition delay time of kerosene under specific temperature, pressure and equivalence ratio can be obtained by solving the ordinary differential equation system of the chemical reaction source term of the model.

Determining an optimization problem: since the number of reactions and the component number of the total package mechanism of kerosene are smaller than those of the skeleton mechanism of kerosene, the ignition delay time calculation of the total package mechanism of kerosene appears to deviate from the calculation result of the more accurate skeleton mechanism. It is necessary to correct the kinetic parameters of the total package mechanism of kerosene so that the calculation of the ignition delay time of the total package mechanism of kerosene can be closer to the calculation result of the skeleton mechanism under a certain thermodynamic state space. In thermodynamic state space is temperature T epsilon [1000, 2000] K, pressure P epsilon [0.5, 10] atm, equivalence ratioAnd uniformly taking 5 multiplied by 4 sample points in a defined thermodynamic state space within the range of E [0.5, 2.0], calculating the average value of the relative error of the ignition delay time between the total packet mechanism and the skeleton mechanism as an optimization objective function, wherein the optimization problem is the minimum value of the objective function.

The objective function is defined as

Wherein the method comprises the steps ofExpressed in temperature/>Pressure/>Equivalence ratio/>The ignition delay time relative error between the kerosene total package mechanism and the skeleton mechanism.

Determining an input space: the fold change of factor a and the fold change of activation energy E before referring to 6 reactions in total of reaction numbers 1,2, 7, 8, 9, and 10 were used as the input spaces in this example. The specific effects of the change times are as follows: the new input parameters are obtained by multiplying 12 kinetic parameters (the kinetic parameters refer to a pre-factor and activation energy) by the respective multiple of change, so that a new kerosene mechanism is formed.

The input spatial dimensions are: the 6 reactions each had 2 fold changes, so the input spatial dimension was 12 dimensions.

The input spatial boundary is: the fold change boundary of the pre-finger factor A is [0.1, 10], and the change boundary of the activation energy E is [0.8, 1.2].

The embodiment is optimized by adopting the optimized dotting method of the scheme and a conventional genetic algorithm (the convergence result of the genetic algorithm can be regarded as a global optimal solution), and the change relation between the optimal value (i.e. the minimum average error) and the sample number under the current sample number can be referred to as fig. 2.

As can be seen from fig. 2, the minimum average error obtained by the genetic algorithm after sampling 37000 samples is 24.64%, while the minimum average error obtained by the present solution after sampling 1915 samples is 24.81%; compared with the prior art, the method can greatly improve the ignition delay time calculation precision of the kerosene total package model under the condition of a small number of samples by adopting the optimization scheme, and the found optimal solution and the global optimal solution have a difference of less than 0.2 percent.

In summary, the sequence dotting optimization method of the scheme has the advantages that a larger initial sample size is not needed, the dotting criterion does not need to depend on a specific proxy model construction method, and the local optimization is not easy to fall into.

Claims (8)

1. The sequence dotting optimization method based on the proxy model is characterized by comprising the following steps:

S1, acquiring an optimization problem of an object to be optimized, determining the dimension and the boundary of an input space based on the optimization problem, and sampling the input space by using a Latin hypercube method to obtain a sample set;

S2, training a proxy model with regression capability by adopting a sample set, calculating a scale factor by adopting an arctangent function, and then selecting a random number between 0 and 1;

S3, judging whether the random number is larger than a scale factor, if so, sampling in an input space by adopting a maximum/small agent model prediction criterion, otherwise, sampling in the input space by adopting a maximum minimum distance criterion;

S4, adding the sampled sample points into a sample set, judging whether the sample set meets a termination condition, if so, outputting a maximum value or a minimum value obtained by the sample set and the agent model, otherwise, entering a step S5;

S5, detecting whether a clustering phenomenon exists in the sample set, if so, entering a step S6, otherwise, directly returning to the step S2;

S6, taking the hypersphere formed by the clustered samples as a current reject domain, adding the hypersphere as a constraint condition into a maximum/small agency model prediction criterion and a minimum distance criterion based on maximization, and returning to the step S2.

2. The proxy model based sequence dotting optimization method of claim 1, wherein the method for detecting whether a cluster phenomenon exists in a sample set comprises:

statistical sample set Any one of samples/>Number M of other samples present nearby:

M=

Wherein, To calculate the number of elements in the collection; /(I)For any sample in the sample set,/>For sample set NOT/>Any sample outside; /(I)Is a 2-norm; /(I)Is the dimension of the input space; /(I)Is the maximum value;

Judgment sample Whether the number M of other samples existing nearby is greater than or equal to a clustering threshold, if so, the samples/>Clustering phenomenon exists nearby; otherwise sample/>No clustering phenomenon exists nearby.

3. The proxy model based sequence dotting optimization method of claim 2 wherein the hyperspheres formed by clustered samples are based on samplesIs the sphere center,/>Is a sphere formed by radius;

The constraint conditions are as follows: Wherein/> Is a sample in the sample set.

4. The proxy model based sequence dotting optimization method of claim 1, wherein the termination condition is that a standard deviation of a predicted optimal solution of the proxy model in a plurality of sequence dotting is less than a standard deviation threshold, and a total sample size is greater than a sample size threshold:

Wherein, Standard deviation for predicting optimal solution; /(I)The optimal solution is predicted for the point adding of the agent model in the current last k round; /(I)The average value of the optimal solution of the prediction of the agent model from the current nth round to the last k round; setting a standard deviation threshold value; /(I) Is the total sample size; /(I)The method comprises the steps of sampling an initial sample size in an input space by using a Latin hypercube method; /(I)For adding number of points,/>Is a sample size threshold; k is a round variable; n is the total number of proxy model updates.

5. The proxy model based sequence dotting optimization method of claim 1, wherein the sample size obtained by sampling in the input space using latin hypercube methodThe method comprises the following steps:

Wherein, For input space,/>And/>Input space/>, respectivelyLower and upper bounds of (2); n is/>Is a dimension of (2); /(I)AndSample/>, respectivelyLower and upper bounds of (2); /(I)Is a proportional symbol; /(I)Is the dimension of the input space.

6. The proxy model based sequence dotting optimization method of any one of claims 1-5, wherein the maximum/minimum proxy model prediction criterion is searching the maximum/minimum proxy model predictions in the whole input space, and the optimization problem is:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) Is an input space.

7. The proxy model based sequence dotting optimization method of any one of claims 1-5, wherein maximizing a minimum distance criterion is to increase input space sample filling rate and to collect a sample setTaking a boundary formed by samples meeting the conditions as a constraint condition, and sampling; the samples satisfying the conditions are:

Wherein x is the sample in the sample set; A predicted value of the agent model for the sample x; /(I) And/>True values/>, respectively, corresponding to samples xMaximum and minimum of (a)/>Is a scale factor; /(I)Samples that meet the condition; /(I)Is a sample set;

The optimization problem of maximizing the minimum distance criterion is:

Wherein, Any sample in the sample set; /(I)And/>Respectively/>Lower bound and upper bound of (2).

8. The proxy model based sequence dotting optimization method of claim 1, wherein the expression of the arctangent function is:

Wherein, Is a scale factor; /(I)Is the sample size threshold.