Complex system performance model uncertainty quantification method based on sPCE

Abstract

The invention relates to an sPCE-based complex system performance model uncertainty quantification method, which can be widely applied to analysis of influence of parameter uncertainty on system performance in a complex system performance model with uncertainty. According to the method, a non-linear function relation between a complex system performance parameter Y and an uncertainty parameter xi is approximately represented through an agent model, and uncertainty quantitative analysis of a complex system performance model is completed. The construction of the proxy model comprises the steps of constructing a performance model of the complex system and determining related input parameters; according to the actually determined uncertainty parameters influencing the performance model, the probability distribution type of each uncertainty parameter is analyzed, and the nonlinear function relation is approximately replaced with chaos polynomial expansion; and determining a mathematical expression of the agent model, determining a multi-dimensional polynomial basis function in the agent model, determining an expansion coefficient of the agent model, and completing agent model construction.

Description

Technical Field

[0001] The present invention belongs to the research field of uncertainty quantification, and specifically relates to an uncertainty quantification method for a complex system performance model based on sPCE, which can be widely used to analyze the impact of parameter uncertainty on system performance in a complex system performance model with uncertainty. Background Art

[0002] Uncertainty quantification refers to the process of quantifying the uncertainty of system output under the internal propagation mechanism of the system based on the uncertainty of system input, external environment and the system itself. It can reasonably consider the impact of uncertainty.

[0003] There are many uncertain factors in the modeling and simulation process, which inevitably leads to deviations between the model prediction results and the experimental data. Therefore, uncertainty quantification is crucial to improve the reliability and robustness of model simulation. In recent years, as systems become increasingly complex, the method of using surrogate models to approximate the original model of complex systems for parameter uncertainty analysis has been widely used. Traditional surrogate model technology has good accuracy and efficiency in solving low-dimensional nonlinear problems. However, with the increase of dimensions, the computational cost of modeling using traditional methods increases greatly. Chaotic polynomial expansion has a rigorous theoretical basis and wide applicability. As a fast-converging and low-cost uncertainty quantification method, it has developed rapidly and become one of the mainstream methods.

[0004] When dealing with high-dimensional and highly coupled problems, the Polynomial Chaos Expansions (PCE) method will inevitably encounter the problem of dimensionality curse, and the computational cost will increase exponentially. Its approximation ability will be greatly reduced, and even a feasible proxy model cannot be obtained. However, complex systems involve high-dimensional uncertainty parameters, which is also the biggest challenge currently faced by using PCE to characterize and quantify uncertainty factors in complex system models. Therefore, it is particularly important to solve the dimensionality curse and improve the convergence speed and computational accuracy of the proxy model when the experimental samples are limited. Summary of the invention

[0005] In view of this, the present invention provides a complex system performance model uncertainty quantification method based on sPCE, and the specific steps of the method are:

[0006] Step 1: Build a performance model of the complex system and determine the relevant input parameters;

[0007] Step 2: Identify the uncertainty parameters that affect the performance model in step 1 ξ=[ξ1,ξ2,...,ξ n], and assume that these n uncertainty parameters are independent of each other, and analyze the probability distribution type of each uncertainty parameter;

[0008] Step 3: The performance parameter of the complex system is Y, and there is a nonlinear function Y = f (ξ) between it and the independent uncertainty parameter. This functional relationship is regarded as a "black box" and the function is approximately replaced by the chaotic polynomial expansion P (ξ), that is:

[0009]

[0010] In the formula: i=(i1,...,i n ) is a multiple subscript and satisfies |i|=i1+...+i n ; Φ i is a multidimensional polynomial basis function.

[0011] Step 4: According to the amount of data and the requirements for accuracy, the cutoff q is determined by user-defined, and the high-order expansion term of equation (1) in step 3 is truncated to obtain the expansion consisting of orthogonal polynomials with an order not exceeding q, which is the q-order generalized chaotic polynomial:

[0012]

[0013] The number of expanded terms is D+1=(q+n)! / (q!n!).

[0014] Step 5: According to the probability distribution form of the uncertainty parameters in step 2, select the orthogonal polynomial corresponding to each uncertainty parameter as the basis function according to Table 1. When the uncertainty parameter cannot be described by the common standard probability distribution type in the table, it can be converted into the required probability distribution type according to the equal probability conversion principle, and the polynomial basis function Φ can be obtained. i (ξ)(i=0,...,D).

[0015] Table 1. Askey solution

[0016] Random distribution form Askey's Law Variable interval normal distribution Hermite polynomials [-∞,∞] Gamma distribution Laguerre Polynomial [0,∞] Beta Distribution Jacobi Polynomials [a,b] Even distribution Legendre polynomials [a,b] Exponential distribution Charlier Polynomial {0,1,2，…} Binomial Distribution Krawtchouk polynomials {0,1,2，…,N} Negative binomial distribution Meixner Polynomials {0,1,2，…} Hypergeometric distribution Hahn polynomials {0,1,2，…,N}

[0017] Step 6: Use Latin hypercube sampling to generate N1 samples of uncertainty parameters, denoted as Execute the function calculation f(ξ) in step 3 for each sample to obtain the performance response Form a training sample set.

[0018] According to the polynomial basis function Φ obtained in step 5 i (ξ) Construct the measurement matrix, which is expressed as follows:

[0019]

[0020] According to equation (1) in step 3 and equation (2) in step 4, the following linear equations can be obtained:

[0021]

[0022] Step 7: Using compressed sensing technology, solve the expansion coefficient to complete the construction of the proxy model. The number of expansion terms in the complete chaotic polynomial increases significantly with the number of uncertainty parameters and the order of the expansion. Only those expansion terms that have a greater impact on the output response can be selected to construct sparse chaotic polynomials (Sparse Polynomial ChaosExpansions, sPCE). The expansion coefficient solution can be expressed as the following optimization problem:

[0023]

[0024] In the formula, the norm ||a||0 represents the number of non-zero elements in vector a, and finally the sparse vector of the polynomial is obtained The length of this vector is D′+1, then the sparse chaotic polynomial, that is, the proxy model of the complex system is The number of polynomials is D′+1.

[0025] Step 8: Use Latin hypercube sampling to generate N2 uncertainty parameter samples and input them into the proxy model obtained in step 7 The uncertainty of the performance of complex systems is quantitatively analyzed in the paper. The random probability characteristics of the performance output response can be calculated by the following formula, that is, the mean μ P and variance σ P .

[0026]

[0027] The advantages and positive effects of the present invention are:

[0028] (1) The present invention proposes a method for quantifying the uncertainty of complex system performance based on chaotic polynomial expansion, which approximates the functional performance model of the complex system into an expanded form of a series of random parameters, thereby quickly obtaining the probabilistic statistical characteristics of the performance response index through a smaller number of calculations, which can greatly reduce the calculation cost and obtain a fairly accurate performance response index.

[0029] (2) The present invention fully considers the complexity and diversity of the distribution types of uncertainty parameters of complex systems. By constructing uncertainty transformation, many distribution types that do not belong to or are far different from the Askey scheme are converted into a certain type of standard distribution type shown in the Askey scheme, such as lognormal distribution and Weibull distribution, so as to find the corresponding polynomial basis functions and construct a proxy model of the complex system.

[0030] (3) The present invention has strong engineering operability. Under engineering constraints such as limited samples and test time, it can address the characteristics of complex systems such as high nonlinearity, multiple inputs and multiple outputs, and strong coupling. A sparse PCE expansion method based on compressed sensing is proposed to solve the problem of dimensionality curse. Through the compressed sensing algorithm, PC bases with little or no influence are eliminated, thereby achieving sparse representation of PCE. In the case of limited samples, an accurate proxy model can be obtained and the calculation time can be greatly shortened.

[0031] (4) The error between the final constructed proxy model and the complex system performance model is less than 10 -5 . BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is a flow chart of the method of the present invention;

[0033] Figure 2 It is a complex system performance analysis model in the implementation case of the present invention; DETAILED DESCRIPTION

[0034] In order to express the concept of the present invention more clearly and intuitively, the present invention is described in detail below with reference to the accompanying drawings and implementation examples.

[0035] like Figure 1 As shown, the present invention provides a complex system performance model uncertainty quantification method based on sPCE, and describes the specific steps of the method in combination with principles and data:

[0036] Step 1: Construct a performance model of a complex system and determine relevant input parameters. This invention takes the performance model of a complex system as an example. Figure 2 As shown, the performance model includes sub-model 1, sub-model 2, sub-model 3, sub-model 4 and sub-model 5. The input parameters of the performance model include 6 parameters, namely parameter 1, parameter 2, parameter 3, parameter 4, parameter 5 and parameter 6, which are used as the input of sub-model 1. Figure 2 The arrows in represent the parameter interactions between the sub-models. For example, the output parameters of sub-model 1 are parameter 7 and parameter 8, and parameter 7 is the input parameter of sub-model 2, and parameter 8 is the input parameter of sub-model 4; the output parameter of the performance model is performance parameter Y. The related input parameters include parameters 1, parameter 2, parameter 3, parameter 4, parameter 5, and parameter 6 of the performance model;

[0037] Step 2: Identify the uncertainty parameters that affect the performance model in step 1 ξ=[ξ1,ξ2,...,ξ n], and it is assumed that these n uncertainty parameters are independent of each other, and the probability distribution type of each uncertainty parameter is analyzed. The uncertainty of complex system performance mainly comes from the five uncertainty parameters of parameter 1, parameter 2, parameter 3, parameter 4 and parameter 5 (parameter 6 is a deterministic parameter), that is, n = 5. The description of uncertainty parameters and probability distribution types are shown in Table 2.

[0038] Table 2 Description of uncertainty parameters

[0039] Uncertainty parameters symbol Parameter code Distribution Type Parameter 1 ΔJ <![CDATA[ξ1]]> normal distribution Parameter 2 ΔC <![CDATA[ξ2]]> Lognormal distribution Parameter 3 Δm <![CDATA[ξ3]]> Even distribution Parameter 4 ΔX <![CDATA[ξ4]]> Gamma distribution Parameter 5 ΔR <![CDATA[ξ5]]> normal distribution

[0040] Step 3: The performance parameter of the complex system is Y, and there is a nonlinear function Y = f (ξ) between it and the independent uncertainty parameter. This functional relationship is regarded as a "black box" and the function is approximately replaced by the chaotic polynomial expansion P (ξ), that is:

[0041]

[0042] Where: a i is the expansion coefficient; i=(i1,...,i n ) is a multiple subscript and satisfies |i|=i1+...+i n ; Φ i is a multidimensional polynomial basis function. Then the expressions of the complex system performance model and uncertainty parameters in this case are:

[0043] Y=f(ξ1,ξ2,ξ3,ξ4,ξ5)

[0044] Among them, ξ1, ξ2, ξ3, ξ4, ξ5 are the five uncertainty parameters affecting the performance of the complex system determined in step 2.

[0045] Step 4: Determine the cutoff q, and truncate the high-order expansion terms of equation (1) in step 3, so as to obtain an expansion consisting of orthogonal polynomials with an order not exceeding q, which is the q-order generalized chaotic polynomial:

[0046]

[0047] Among them, the number of expansion terms is D+1=(q+n)! / (q!n!), which is determined by the number of uncertainty parameters n and the order q of the polynomial. In this case, under the premise of satisfying the calculation accuracy, considering the calculation cost, equation (1) is truncated to order q, taking q=3, n=5 (see step 2), then the number of terms in the chaotic polynomial expansion is D+1=(q+n)! / (q!n!)=(3+5)! / (3!5!)=56, then D=55, ​​and the chaotic polynomial expansion after truncation, that is, the proxy model can be expressed as:

[0048]

[0049] Step 5: According to the probability distribution form of the uncertainty parameters in step 2, select the orthogonal polynomial corresponding to each uncertainty parameter as the basis function according to Table 1. When the uncertainty parameter cannot be described by the common standard probability distribution type in the table, it can be converted into the required probability distribution type according to the equal probability conversion principle. In this case, since ξ2 obeys the lognormal distribution (see Table 2), the corresponding basis function cannot be found in the Askey scheme, so the logarithmic transformation is selected to convert it into a normal distribution, that is:

[0050] ξ2′=lgξ2

[0051] According to the Askey scheme, the corresponding basis function is selected according to the distribution type of the uncertainty parameter determined in step 2, as shown in Table 3.

[0052] Table 3. Basis function table corresponding to the five uncertainty parameters

[0053]

[0054]

[0055] Then the chaotic polynomial P q (ξ) Expanded basis function Φ i (ξ)(i=0,...,55) are 1, 17.3·ξ1, 9.6·ξ2, 0.04·ξ3, 0.07·ξ4, 10.2·ξ4,..., 0.001·ξ3 respectively 3 -0.1·ξ3,6614·ξ1 3 -39.7·ξ1, 56 items in total.

[0056] Step 6: Use Latin hypercube sampling to generate N1 samples of uncertainty parameters, denoted as Execute the function calculation f(ξ) for each sample and obtain the performance response In this case, 280 sample points are extracted within the definition domain of the 5-dimensional uncertainty parameter by Latin hypercube sampling, denoted as Substitute these samples into the performance function of the complex system to calculate the corresponding response value, recorded as Y j (j=1,…,280), forming a training sample set.

[0057] According to the polynomial basis function Φ obtained in step 5 i (ξ) Construct the measurement matrix, which is expressed as follows:

[0058]

[0059] According to equation (1) in step 3 and equation (2) in step 4, the following linear equations can be obtained:

[0060]

[0061] Step 7: Use compressed sensing technology to solve the expansion coefficient The number of expansion terms in the complete PCE increases significantly with the number of uncertainty parameters and the order of the expansion. Only those expansion terms that have a greater impact on the output response can be selected to construct the sparse chaotic polynomial. The expansion coefficient solution can be expressed as the following optimization problem:

[0062]

[0063] Here, the norm ||a||0 represents the number of non-zero elements in vector a.

[0064] The orthogonal matching pursuit method can be used to solve the above optimization problem, and a sparse solution of the development coefficient vector can be obtained under a limited number of sampling samples. The steps of the orthogonal matching pursuit algorithm are as follows:

[0065] Input: Performance response P∈R 280 , the measurement matrix Θ∈R 280×56 .

[0066] Output: sparse coefficient vector a∈R 56 .

[0067] Initialization: residual vector γ0 = P q , coefficient vector a=0, index set Λ0=φ, let t=0

[0068] Step 1: When ||γ t ||2>10 -5 &t<400, continue execution;

[0069] Step 2: The jth t Column and residual vector γ t-1 When the correlation is strongest, Let Λ t =Λ t-1 ∪j t ;

[0070] Step 3: Determine the coefficient a corresponding to the currently selected column by the least squares method t =argmin x ||P q -Θ Λt a||2;

[0071] Step 4: Update the residual γ t =P q -θa t , and let t = t + 1;

[0072] Repeat steps 1 to 4 until the iteration stop criterion in step 1 is met; output the result.

[0073] In this case, the input is the sample training set P(ξ (j) ),j=1,…,280 and the 280*56 measurement matrix Θ formed by the basis function, and the final vector This is the expansion coefficient of the desired sparse chaotic polynomial. At this time, the number of expansion terms is D'+1=17. The proxy model of the complex system performance model obtained is:

[0074]

[0075] Step 8: Use Latin hypercube sampling to generate N2 = 300 uncertainty samples and input them into the proxy model obtained in step 7 The uncertainty of the performance of complex systems is quantitatively analyzed in the paper. The random probability characteristics of the performance output response can be calculated by the following formula, that is, the mean μ P and variance σ P .

[0076]

[0077] The implementation case of the present invention takes a complex system performance analysis model as an example. Figure 2 As shown in the figure, the uncertainty quantification method of complex system performance model based on sPCE is only explained in detail from the data perspective.

[0078] According to step 1: construct a performance model of the complex system, and determine that the input parameters of the performance model of the complex system include parameter 1, parameter 2, parameter 3, parameter 4, parameter 5 and parameter 6;

[0079] According to step 2: the uncertainty of the complex system performance is analyzed to be mainly derived from the five uncertainty parameters of parameter 1, parameter 2, parameter 3, parameter 4 and parameter 5 (parameter 6 is a deterministic parameter), that is, n = 5. The description of the uncertainty parameters and the probability distribution type are shown in Table 2.

[0080] According to step 3: The expressions of the complex system performance model and uncertainty parameters in this case are:

[0081] Y=f(ξ1,ξ2,ξ3,ξ4,ξ5)

[0082] Among them, ξ1, ξ2, ξ3, ξ4, and ξ5 are the five main uncertainty parameter vectors that affect the performance of the complex system.

[0083] According to step 4: In this case, under the premise of satisfying the calculation accuracy, considering the calculation cost, take q = 3, n = 5 (see step 2), then the number of terms of the chaotic polynomial expansion is D + 1 = (q + n)! / (q! n!) = (3 + 5)! / (3! 5!) = 56, then D = 55, the truncated chaotic polynomial expansion, that is, the proxy model can be expressed as:

[0084]

[0085] According to step 5: According to the probability distribution form of the random parameters and the Askey scheme in Table 1, the corresponding orthogonal polynomial is selected as the basis function. In this case, since ξ2 obeys the log-normal distribution (see Table 2), the corresponding basis function cannot be found in the Askey scheme, so the logarithmic transformation is selected to convert it into a normal distribution, that is:

[0086] ξ2′=lgξ2

[0087] According to the Askey scheme, the corresponding basis function is selected for the distribution characteristics of the uncertainty parameters determined in step 2, as shown in Table 3. Then the basis function Φ of the chaotic polynomial P3(ξ) i (ξ) are 1, 17.3·ξ1, 0.04·ξ3, 0.07·ξ4, 10.2·ξ4,…, 0.001·ξ3 3 -0.1·ξ3,6614·ξ1 3 -39.7·ξ1, 56 items in total.

[0088] According to step 6: 280 sample points are extracted within the domain of the 5-dimensional input variable through Latin hypercube sampling, denoted as Substitute these sample values ​​into the complex system performance simulation model to calculate the corresponding response value, denoted as Y j (j=1,…,280), forming a training sample set.

[0089] According to the polynomial basis function Φ obtained in step 5 i (ξ) Construct the measurement matrix, which is expressed as follows:

[0090]

[0091] According to equation (1) in step 3 and equation (2) in step 4, the following linear equations can be obtained:

[0092]

[0093] According to step 7: Use compressed sensing technology to solve the expansion coefficient The proxy model of complex system performance based on sparse chaotic polynomial expansion (sPCE) is obtained as follows:

[0094]

[0095] According to step 8: Use Latin hypercube sampling method to generate 300 random parameter samples, input them into the proxy model P3(ξ) obtained in step 7 to perform uncertainty quantitative analysis on the performance of the complex system, and obtain the mean value μ of the performance P =10.34, variance σ P 2 =0.066.

Claims

1. A method for quantifying uncertainty in complex system performance models based on sPCE, characterized by: The nonlinear functional relationship between the complex system performance parameter Y and the uncertainty parameter ξ is approximately represented by the surrogate model, and the uncertainty quantitative analysis of the complex system performance model is completed.

2. A complex system performance model uncertainty quantification method based on sPCE according to claim 1, characterized in that: The proxy model construction process is as follows: Step 1: Build a performance model of the complex system and determine the relevant input parameters; Step 2: Determine the uncertainty parameter ξ=[ξ1,ξ2,...,ξ n ], analyzing the probability distribution type of each uncertainty parameter, wherein the uncertainty parameters are independent of each other; Step 3: There is a nonlinear functional relationship Y=f(ξ) between the performance parameter Y of the complex system and the uncertainty parameter ξ, and the nonlinear functional relationship Y=f(ξ) is approximately replaced by the chaotic polynomial expansion P(ξ); Step 4: Determine the mathematical expression of the proxy model. Specifically, customize the cutoff q and perform high-order expansion term cutoff on P(ξ) to obtain the mathematical expression P of the proxy model. q (ξ), expressed as follows In the formula, a is the expansion coefficient, which is i Represents; Θ is the measurement matrix, which is composed of multidimensional polynomial basis functions Φ i Indicates that the number of expanded items is D+1=(q+n)! / (q!n!); Step 5: Determine the multidimensional polynomial basis function Φ in the surrogate model i ; Step 6: Determine the expansion coefficient of the proxy model and complete the construction of the proxy model.

3. A complex system performance model uncertainty quantification method based on sPCE according to claim 2, characterized in that: The multidimensional polynomial basis function Φ i The method for determining is to determine the multidimensional polynomial basis function Φ according to the probability distribution type of the uncertainty parameter based on the Askey scheme. i When the uncertainty parameter cannot be described by the standard probability distribution type, it is converted into the required probability distribution type according to the equal probability conversion principle to obtain the polynomial basis function Φ i (ξ).

4. A complex system performance model uncertainty quantification method based on sPCE according to claim 3, characterized in that: The method for determining the expansion coefficient of the proxy model is as follows: The Latin hypercube sampling method is used to generate samples of N1 uncertainty parameters, denoted as Execute Y = f(ξ) for each sample and obtain the performance response Forming a training sample set; According to the obtained polynomial basis function Φ i (ξ) Construct the measurement matrix Θ, which is expressed as follows: Using the obtained performance response Y, the measurement matrix Θ, and P q (ξ) = Θa. Using compressed sensing technology, the expansion coefficient a is estimated and the proxy model is constructed.

5. A method for quantifying uncertainty of complex system performance models based on sPCE according to any one of claims 1 to 4, characterized in that: The Latin hypercube sampling method is used to generate N2 uncertainty parameter samples, which are input into the proxy model to complete the uncertainty quantitative analysis of the performance of the complex system.

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