Data processing method, device, medium and product based on polynomial chaos expansion

Abstract

The present invention discloses a data processing method, device, medium, and product based on polynomial chaotic expansion. The method obtains several sample points of a random variable as a sample data set; performs polynomial chaotic expansion using orthogonal polynomial basis functions and solves for the expansion coefficients in the polynomial chaotic expansion; trains several generative models to learn the distribution of the absolute value of the normalized Galerkin projection inner product term; updates the probability density function according to the generative model, obtains several new sample points of the random variable based on the updated probability density function, and incorporates the new sample points into the sample data set; re-solves for the expansion coefficients in the polynomial chaotic expansion until an iterative termination condition is met, and generates a polynomial chaotic expansion proxy model based on the last solved expansion coefficients. The present invention can improve the fitting accuracy of the proxy model, effectively reduce the amount of computation, and improve data processing efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of machine learning models, and in particular to a data processing method, device, medium and product based on polynomial chaos expansion. Background Art

[0002] Uncertainty quantification (UQ) is a crucial research area in engineering and applied science. Polynomial Chaos Expansion (PCE), a key UQ method, has been widely applied to various uncertainty quantification problems due to its universality. The core idea of ​​PCE is to construct a computationally efficient proxy model using orthogonal polynomial basis functions to approximate the complex original computational model.

[0003] This modeling approach provides an effective tool for studying complex problems in fields such as engineering, finance, and the natural sciences. For example, in the engineering field, simply supported beams are widely used as basic components, and their research is crucial to the smooth implementation of various engineering projects. By in-depth research on simply supported beams, engineers can better understand and design various structural systems, thereby ensuring the safety, economy, and stability of the structures. In the scenario of studying the center displacement of a simply supported beam when subjected to uncertain external forces (elastic perturbations), generally speaking, the external force borne by the simply supported beam, the length of the beam, the elastic modulus of the beam, and the cross-sectional moment of inertia of the beam are all random variables and satisfy certain given random distributions. According to the classical elastic formula, the center displacement of the beam is also a random variable at this time. The method based on polynomial chaos expansion can quantify the uncertainty of the center displacement of the simply supported beam.

[0004] However, the inventors discovered that existing techniques suffer from at least the following issues: When applying existing polynomial chaos expansion algorithms to practical problems, the sample points of random variables obtained using least squares sampling can deviate from the true model during calculations, leading to errors in the calculated results. Therefore, achieving a good fit for areas with significant variation in the model is crucial, and solving similar problems is currently in demand in many important fields such as engineering, finance, and the natural sciences. Summary of the Invention

[0005] The purpose of the embodiments of the present invention is to provide a data processing method, device, medium and product based on polynomial chaos expansion, which can effectively improve the fitting accuracy of the polynomial chaos expansion agent model, effectively reduce the amount of calculation, and improve data processing efficiency.

[0006] To achieve the above objectives, an embodiment of the present invention provides a data processing method based on polynomial chaos expansion, comprising:

[0007] Determine the original calculation model of the parameters to be solved and the random variables that meet the preset probability density function;

[0008] Obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set;

[0009] According to the sample data set, the original calculation model is subjected to polynomial chaos expansion using orthogonal polynomial basis functions, and expansion coefficients in the polynomial chaos expansion are solved;

[0010] Training a plurality of generative models; wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions;

[0011] updating the probability density function according to the generative model, obtaining a plurality of newly added sample points of the random variable according to the updated probability density function, and incorporating the newly added sample points into the sample data set;

[0012] Re-execute step: according to the sample data set, use the preset orthogonal polynomial basis function to perform polynomial chaos expansion on the original calculation model, and solve the expansion coefficient in the polynomial chaos expansion until the preset iteration termination condition is met, and generate the polynomial chaos expansion proxy model of the parameter to be solved according to the expansion coefficient obtained by the last solution.

[0013] As an improvement to the above solution, the method of performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set and solving the expansion coefficients in the polynomial chaos expansion includes:

[0014] Obtaining an actual observation value corresponding to each sample point in the sample data set;

[0015] According to each of the sample points, a preset orthogonal polynomial basis function is used to perform a polynomial chaotic expansion on the original calculation model to obtain a polynomial chaotic expansion containing the expansion coefficient to be solved;

[0016] The expansion coefficient is calculated by minimizing the residual sum of squares by the least squares method; wherein the residual sum of squares is obtained by summing the squares of the difference between the polynomial chaotic expansion of each sample point and the corresponding actual observation value.

[0017] As an improvement to the above solution, the training of several generative models, wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions, include:

[0018] Calculating the distribution of absolute values ​​of Galerkin projection inner product terms of the orthogonal polynomial basis functions;

[0019] Training several generative models to learn the distribution of the normalized absolute value of the Galerkin projection inner product term;

[0020] The model parameters of the generative model are optimized by minimizing the KL divergence.

[0021] As an improvement to the above solution, obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set includes:

[0022] According to the probability density function, a number of sample points of the random variable are obtained by uniform sampling or Latin hypercube sampling as a sample data set.

[0023] As an improvement to the above solution, the preset iteration termination condition is:

[0024] The number of adaptive iterations reaches a preset iteration threshold;

[0025] or,

[0026] The residual sum of squares is less than a preset error threshold, and the residual sum of squares is obtained by summing the squares of the differences between the polynomial chaotic expansion of each sample point and the corresponding actual observation value.

[0027] As an improvement of the above solution, the parameter to be solved is the center displacement of the simply supported beam; the random variables include the external force acting on the simply supported beam, the length, elastic modulus and section inertia moment of the simply supported beam.

[0028] An embodiment of the present invention further provides a data processing device based on polynomial chaotic expansion, comprising:

[0029] A random variable determination module is used to determine the original calculation model of the parameters to be solved and the random variables that meet the preset probability density function;

[0030] A sample data set generation module, configured to obtain a plurality of sample points of the random variable according to the probability density function as a sample data set;

[0031] A polynomial chaos expansion module is used to perform polynomial chaos expansion on the original calculation model according to the sample data set using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion;

[0032] A generative model training module, configured to train a plurality of generative models; wherein the generative models are configured to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions;

[0033] a sample data set updating module, configured to update the probability density function according to the generative model, obtain a plurality of newly added sample points of the random variable according to the updated probability density function, and incorporate the newly added sample points into the sample data set; and trigger the polynomial chaos expansion module to re-execute the steps of: performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion;

[0034] The proxy model generation module is used to generate a polynomial chaos expansion proxy model of the parameter to be solved according to the expansion coefficient obtained at the last solution when a preset iteration termination condition is met.

[0035] An embodiment of the present invention also provides a data processing device based on polynomial chaotic expansion, comprising a processor, a memory, and a computer program stored in the memory and configured to be executed by the processor. When the processor executes the computer program, it implements the data processing method based on polynomial chaotic expansion as described in any one of the above.

[0036] An embodiment of the present invention also provides a computer-readable storage medium, which includes a stored computer program, wherein when the computer program is running, the device where the computer-readable storage medium is located is controlled to execute the data processing method based on polynomial chaos expansion as described in any one of the above.

[0037] An embodiment of the present invention further provides a computer program product, which includes a computer program or computer instructions. When the computer program or the computer instructions are executed by a processor, the data processing method based on polynomial chaos expansion as described in any one of the above is implemented.

[0038] Compared with the prior art, the data processing method, device, medium and product based on polynomial chaos expansion disclosed in the present invention adopt the technical means of the embodiments of the present invention, and realize data processing based on the polynomial chaos expansion method of adaptive importance sampling. The generative model is used to learn the absolute value distribution of the normalized Galerkin projection inner product term, and it is possible to construct an approximate probability density function in the random parameter space and efficiently extract the required samples directly from the generative model. By iteratively and adaptively generating new sample points and dynamically integrating them into the overall sample set, and then solving the undetermined coefficients of the polynomial chaos expansion, the quality of the collected samples can be effectively improved, thereby reducing the number of samples collected. While saving costs, due to the reduction in the sample size, the calculation cost of the expansion coefficient is also reduced accordingly, which significantly improves the fitting accuracy and convergence efficiency of the proxy model. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 This is a flow chart of a data processing method based on polynomial chaos expansion provided by an embodiment of the present invention;

[0040] Figure 2 1 is a flow chart of a preferred data processing method based on polynomial chaos expansion in an embodiment of the present invention;

[0041] Figure 3 This is a schematic diagram of the principle of uniform sampling;

[0042] Figure 4 Schematic diagram of the principle of importance sampling in an embodiment of the present invention;

[0043] Figure 5 is a schematic diagram of the distribution characteristics of random variables in an embodiment of the present invention;

[0044] Figure 6 Schematic diagram of the distribution of sample sampling points in an embodiment of the present invention;

[0045] Figure 7 It is a structural diagram of a data processing device based on polynomial chaos expansion provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0046] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0047] In the description of this application, it should be understood that the terms "center", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings, and are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on this application.

[0048] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of the technical features being referred to. Thus, a feature specified as "first" or "second" may explicitly or implicitly include one or more of such features. Throughout this application, unless otherwise specified, "plurality" means two or more.

[0049] In the description of this application, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they can refer to fixed connections, detachable connections, or integral connections; mechanical connections or electrical connections; direct connections or indirect connections through an intermediate medium; and internal connections between two components. Those skilled in the art will understand the specific meanings of the above terms in this application based on the specific circumstances.

[0050] See also Figure 1 , is a flow chart of a data processing method based on polynomial chaos expansion provided by an embodiment of the present invention. The embodiment of the present invention provides a data processing method based on polynomial chaos expansion, including steps S11 to S16:

[0051] S11, determining an original calculation model of the parameter to be solved and a random variable that satisfies a preset probability density function;

[0052] S12. Obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set;

[0053] S13. Performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion;

[0054] S14, training a plurality of generative models; wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions;

[0055] S15. Update the probability density function according to the generation model, obtain a plurality of new sample points of the random variable according to the updated probability density function, and incorporate the new sample points into the sample data set;

[0056] S16. Re-execute step: according to the sample data set, use a preset orthogonal polynomial basis function to perform a polynomial chaos expansion on the original calculation model, and solve the expansion coefficient in the polynomial chaos expansion until the preset iteration termination condition is met, and generate a polynomial chaos expansion proxy model of the parameter to be solved according to the expansion coefficient obtained by the last solution.

[0057] It should be noted that the polynomial chaos expansion technology uses orthogonal polynomial basis functions to construct a computationally efficient proxy model to approximate the complex original computational model. Consider using polynomial chaos expansion to approximate a finite variance original computational model. . It is assumed that is a random input variable and it obeys the probability density function , then the following polynomial chaos expansion can be used to approximate the original computational model:

[0058]

[0059] in, According to the probability density function Or multivariate orthogonal polynomials selected by data, which are obtained by tensor product of corresponding univariate orthogonal polynomials; is the undetermined expansion coefficient that needs to be solved, usually fitted by existing data and calculated using the least squares method; N is the truncation order, which controls the complexity of the model. When the data used is strictly in accordance with the probability density function Sampling is performed, and when When the least square method is used to calculate will tend to the following Galerkin projection, namely:

[0060]

[0061] However, in many practical problems the probability density function It is unknown, which leads to deviation from the true model during the sampling process of sample points, such as when using the sample points obtained by usual uniform distribution or Latin hypercube sampling for calculation, resulting in errors.

[0062] To solve the above problems, the embodiment of the present invention adopts an adaptive importance sampling method to iteratively and adaptively generate high-quality sample points, so that the areas with relatively large changes in the model can be better fitted.

[0063] In the embodiment of the present invention, the original calculation model of the parameters to be solved is determined according to the actual problem. , and the input value of the original calculation model, as a random variable, the random variable obeys the probability density function The distribution characteristics of is unknown, so it is assumed to be uniformly distributed here.

[0064] According to the probability density function , use the preset sampling algorithm to sample and obtain n sample points of the random variable .

[0065] Preferably, step S12, i.e., obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set, includes:

[0066] According to the probability density function, a number of sample points of the random variable are obtained by uniform sampling or Latin hypercube sampling as a sample data set.

[0067] For each of the sample points , select the appropriate orthogonal polynomial basis function and write the polynomial chaos expansion of the original computational model, that is:

[0068]

[0069] Solve the expansion coefficients in the chaotic expansion of the polynomial It should be noted that existing methods, such as the least squares method, can be used here to solve the expansion coefficients , a polynomial chaos expansion agent model of general accuracy is obtained. The specific means can refer to the existing technology and will not be described here.

[0070] Furthermore, using N generative models Learning the normalized Galerkin projection inner product of the orthogonal polynomial basis function The distribution of , you can choose a generative model such as Normalizing Flow (NF), and learn to approximate it through the parameter θ The distribution of , accurately captures the area where the function changes dramatically, and guides importance sampling.

[0071] According to N generative models Calculate the distribution of the data set and assign it to the probability density function , realize the probability density function Then according to the updated probability density function Resample to obtain several new sample points of the random variable, and update The probability density is higher near the failure boundary, so new samples will appear densely in this area. These newly added sample points can better capture the part of the function with larger changes. The newly added sample points are added to the sample data set to merge into a new sample data set.

[0072] Then jump to step S13 and re-execute the steps: according to the sample data set, use the preset orthogonal polynomial basis function to perform polynomial chaos expansion on the original calculation model, and solve the expansion coefficients in the polynomial chaos expansion.

[0073] The iterative calculation is carried out until the preset iteration termination condition is met, and the expansion coefficient obtained by the final solution is Generate a polynomial chaos expansion proxy model of the parameters to be solved , which is used to replace the complex original calculation model in the subsequent application process to realize the calculation of the parameters to be solved.

[0074] The technical means of the embodiment of the present invention are adopted to realize data processing based on the polynomial chaos expansion method of adaptive importance sampling, and the generative model is used to learn the absolute value distribution of the normalized Galerkin projection inner product term, so as to construct an approximate probability density function in the random parameter space and efficiently extract the required samples directly from the generative model. By iteratively and adaptively generating new sample points and dynamically integrating them into the overall sample set, and then solving the undetermined coefficients of the polynomial chaos expansion, the quality of the collected samples can be effectively improved, thereby reducing the number of samples collected. While saving costs, due to the reduction in the sample size, the calculation cost of the expansion coefficient is also reduced accordingly, which significantly improves the fitting accuracy and convergence efficiency of the proxy model.

[0075] As a preferred embodiment, the embodiment of the present invention is further implemented on the basis of the above embodiment, step S13, that is, according to the sample data set, using orthogonal polynomial basis functions to perform polynomial chaos expansion on the original calculation model, and solving the expansion coefficients in the polynomial chaos expansion, including steps S131 to S133:

[0076] S131, obtaining the actual observation value corresponding to each sample point in the sample data set;

[0077] S132. Performing a polynomial chaotic expansion on the original calculation model using a preset orthogonal polynomial basis function according to each sample point to obtain a polynomial chaotic expansion containing the expansion coefficient to be solved;

[0078] S133. Minimize the residual sum of squares by the least squares method to calculate the value of the expansion coefficient; wherein the residual sum of squares is obtained by summing the squares of the difference between the polynomial chaotic expansion of each sample point and the corresponding actual observation value.

[0079] In the embodiment of the present invention, after sampling to obtain a number of sample points, the original calculation model can be used to directly calculate, or actually measure or observe, or obtain the actual observation value corresponding to each sample point based on prior knowledge. , choose a suitable orthogonal polynomial basis to write the polynomial chaos expansion, and estimate the expansion coefficient by minimizing the following residual square sum by the least squares method:

[0080]

[0081] If the matrix for:

[0082]

[0083] The undetermined coefficient vector can be obtained as:

[0084]

[0085] At this point, a general-precision polynomial chaos expansion proxy model is obtained, and the undetermined coefficients can be approximately expressed as:

[0086]

[0087] The embodiment of the present invention solves the expansion coefficients of the polynomial chaotic expansion by the least square method, which can effectively avoid the limitation of sampling at grid points of the sparse grid and can flexibly process high-dimensional data.

[0088] As a preferred embodiment, the embodiment of the present invention is further implemented on the basis of the above embodiment, step S14, that is, the training of a plurality of generative models; wherein the generative model is used to learn the distribution of the absolute value of the inner product term of the Galerkin projection of the normalized orthogonal polynomial basis function, including S141 to S143:

[0089] S141, calculating the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the orthogonal polynomial basis functions;

[0090] S142. Training a plurality of generative models to learn the distribution of the normalized absolute value of the Galerkin projection inner product term;

[0091] S143. Optimize the model parameters of the generation model by minimizing the KL divergence.

[0092] In an embodiment of the present invention, the Galerkin projection inner product term of the orthogonal polynomial basis function is:

[0093]

[0094] use Generative Model Learn the distribution of the absolute value of the normalized Galerkin projection inner product term separately, and obtain the parameters in the generative model by minimizing the following KL divergence :

[0095]

[0096] because Usually unknown or high-dimensional and difficult to sample, use As an operational approximation, the embodiment of the present invention uses a generative model such as Normalizing Flow (NF) to learn and approximate the NF is a reversible neural network that can transform simple distributions (such as Gaussian) into complex distributions.

[0097] Thus, N generative models are obtained through training, which are used for subsequent updating of the probability density function and sample collection. Make it close to the current optimal importance distribution, guide the next round of sampling to focus more on key areas, avoid artificially preset distribution (such as Gaussian), and be completely data-driven.

[0098] The technical approach employed in the embodiments of the present invention eliminates the need for a complex a posteriori evaluation process and instead employs a generative model to learn the absolute value distribution of the normalized Galerkin projection inner product term. This allows for the construction of an approximate probability density function in the random parameter space, allowing efficient extraction of the desired samples directly from the generative model. This strategy significantly improves the fitting accuracy and convergence efficiency of the proxy model by iteratively and adaptively generating new sample points and dynamically integrating them into the overall sample set, then employing the least squares method to solve for the undetermined coefficients of the polynomial chaotic expansion.

[0099] As a preferred implementation, the embodiment of the present invention is further implemented on the basis of any of the above embodiments, and the preset iteration termination condition is:

[0100] The number of adaptive iterations reaches a preset iteration threshold;

[0101] or,

[0102] The residual sum of squares is less than a preset error threshold, and the residual sum of squares is obtained by summing the squares of the differences between the polynomial chaotic expansion of each sample point and the corresponding actual observation value.

[0103] See also Figure 2 , is a flow chart of a preferred data processing method based on polynomial chaos expansion in an embodiment of the present invention. In this embodiment of the present invention, a suitable maximum number of adaptive iterations is predetermined as an iteration number threshold, and / or a minimum residual square sum is determined as an error threshold.

[0104] In the process of iteratively adaptively generating sample points and calculating expansion coefficients, if the current adaptive iteration number is greater than the preset iteration number threshold, or the current calculated residual square sum is less than the preset error threshold, it indicates that the iteration termination condition has been met. Generate a polynomial chaos expansion proxy model of the parameters to be solved .

[0105] The following compares the adaptive importance sampling method of the embodiment of the present invention with the traditional uniform sampling method to illustrate the beneficial effects of the embodiment of the present invention.

[0106] See also Figure 3 and Figure 4 , Figure 3 This is a schematic diagram of the principle of uniform sampling. Figure 4 This is a schematic diagram of the principle of importance sampling in an embodiment of the present invention. First, the adaptive importance sampling method used is described. Assume that an integral needs to be calculated. The integrand of this integral is a normal distribution with a mean of 0 and a variance of 0.01, and the integral is calculated using the Monte Carlo method. If sample points obtained from a uniform distribution are used as the data set, the error in calculating the integral will be very large, such as Figure 3 However, if importance sampling is used and the point is sampled from the Gaussian distribution as the data set, the integral accuracy calculated using the Monte Carlo method will be greatly improved, as shown in Figure 4 At the same time, the sample points can better capture the places where the integrand changes greatly, such as Figure 3 and Figure 4 As shown in Figure 2, the samples of importance sampling are almost all distributed in the steep area [-0.5, 0.5].

[0107] The integrand in the embodiment of the present invention is relatively complex. The integrand is , and is not necessarily always positive and may be high-dimensional. In order to calculate more accurate unknown coefficients, the embodiment of the present invention uses a generative model to learn the absolute value of the integrand, and then obtains new sample points from the generative model, so that the resampled new sample points can better capture the parts of each integrand that have relatively large changes.

[0108] See also Figure 5 and Figure 6 , Figure 5 is a schematic diagram of the distribution characteristics of random variables in an embodiment of the present invention, Figure 5 This is a schematic diagram of the distribution of sample points in an embodiment of the present invention, which shows the use of a generative model to learn the absolute value of a two-dimensional integrand. It can be observed that the generative model can better capture the function change of the integrand at the point [0.5, 0.5]. Figure 5 At the same time, the new training points obtained by using the generative model sampling are also roughly distributed in the important areas of function change, such as Figure 6 Therefore, for each individual coefficient , we can obtain some sample points that can make the calculation of the inner product term of the Galerkin projection more accurate by training the corresponding generative model. However, we do not use these sample points to directly calculate the inner product term. Instead, we add these sample points that can better capture the characteristics of the real model function to the original data set, and use the least squares method to calculate the new polynomial chaos expansion alternative model again.

[0109] Using the technical means of the embodiments of the present invention, a polynomial chaos expansion method based on adaptive importance sampling is proposed. This method better captures the characteristics of the real model by rationally and adaptively selecting sample points, thereby more accurately calculating the alternative model. Each time a new sample is adaptively generated, it is added to the original data set and calculated together, which has high data utilization and no wasted computation. By selecting a reasonable inner product term for learning, the statistics of the model, such as the mean and variance, can be efficiently calculated using the expansion coefficients.

[0110] The proposed method has the following advantages: First, samples used in the error assessment process can be recycled, effectively improving computational resource utilization; second, the adaptive process employs an incremental sample point addition strategy, fully preserving historical sample data; and finally, the accuracy requirements for the generative model required for sampling are relatively relaxed, further reducing computational overhead. This method offers significant advantages in computing statistics (such as mean and variance) of polynomial chaotic expansions. By focusing on learning the absolute value probability distribution of the inner product term of the Galerkin projection corresponding to the target statistic, it effectively improves the accuracy of statistical calculations.

[0111] As a preferred embodiment, an embodiment of the present invention is further implemented on the basis of any of the above embodiments. In a specific implementation scenario, the parameter to be solved is the center displacement of the simply supported beam; the random variable is at least one of the external force acting on the simply supported beam, the length of the simply supported beam, the elastic modulus and the section moment of inertia.

[0112] In another specific implementation scenario, the parameter to be solved is a cloud feedback parameter, and the random variable includes cloud albedo and / or aerosol concentration.

[0113] It can be understood that the above scenario is only an optional specific implementation method. In actual application, the data processing method based on polynomial chaos expansion of the embodiment of the present invention can also be applied to other fields, such as: computational fluid dynamics, solving the problem of output uncertainty caused by random parameters in the fluid model (such as viscosity, boundary conditions), thereby quickly quantifying the statistical characteristics of flow field variables (velocity, pressure), replacing expensive direct numerical simulation; or realizing climate and earth system modeling, solving the uncertainty of climate model parameters (such as cloud albedo) affecting the prediction results, thereby calibrating the parameter distribution and quantifying the confidence interval of the prediction results; or performing electronic circuit design, solving the problem of circuit performance fluctuations caused by device manufacturing tolerances, and analyzing the impact of random parameters on signal integrity and power consumption, etc.

[0114] For example, taking the structural reliability analysis scenario of a simply supported beam as an example, an embodiment of the present invention provides a data processing method based on polynomial chaos expansion, the method comprising steps S21 to S26:

[0115] S21. Obtaining the external force acting on the simply supported beam, the length, elastic modulus, and section moment of inertia of the simply supported beam as random variables, and using the center displacement of the simply supported beam as a parameter to be solved, to obtain an original calculation model for the center displacement of the simply supported beam; wherein the random variables satisfy a preset probability density function;

[0116] S22. Obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set;

[0117] S23. Performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion;

[0118] S24, training a plurality of generative models; wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions;

[0119] S25. Update the probability density function according to the generation model, obtain a plurality of new sample points of the random variable according to the updated probability density function, and incorporate the new sample points into the sample data set;

[0120] S26. Re-execute step: according to the sample data set, use a preset orthogonal polynomial basis function to perform a polynomial chaos expansion on the original calculation model, and solve the expansion coefficient in the polynomial chaos expansion until the preset iteration termination condition is met, and generate a polynomial chaos expansion proxy model of the center displacement of the simply supported beam according to the expansion coefficient finally solved.

[0121] Preferably, performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set and solving the expansion coefficients in the polynomial chaos expansion includes:

[0122] Obtaining an actual observation value corresponding to each sample point in the sample data set;

[0123] According to each of the sample points, a preset orthogonal polynomial basis function is used to perform a polynomial chaotic expansion on the original calculation model to obtain a polynomial chaotic expansion containing the expansion coefficient to be solved;

[0124] The expansion coefficient is calculated by minimizing the residual sum of squares by the least squares method; wherein the residual sum of squares is obtained by summing the squares of the difference between the polynomial chaotic expansion of each sample point and the corresponding actual observation value.

[0125] Preferably, the training comprises several generative models, wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions, including:

[0126] Calculating the distribution of absolute values ​​of Galerkin projection inner product terms of the orthogonal polynomial basis functions;

[0127] Training several generative models to learn the distribution of the normalized absolute value of the Galerkin projection inner product term;

[0128] The model parameters of the generative model are optimized by minimizing the KL divergence.

[0129] Preferably, the preset iteration termination condition is: the number of adaptive iterations reaches a preset iteration threshold; or the sum of squared residuals is less than a preset error threshold.

[0130] In the embodiment of the present invention, the data processing method based on polynomial chaos expansion can be applied in the field of structural reliability analysis (engineering). In the simply supported beam problem, assuming is the external force acting on the beam, is the length of the beam, is the elastic modulus of the material, is the section moment of inertia of the beam, then the center displacement of the beam (elastic disturbance) is:

[0131]

[0132] Further assume that the external force Satisfying the mean , the standard deviation is Normal distribution, the length of the beam Satisfying the mean , the standard deviation is Normal distribution, elastic modulus Satisfying the mean , the standard deviation is Normal distribution, section moment of inertia Satisfying the mean , the standard deviation is The normal distribution of . At this time, the center displacement It is also a random variable. If we denote the random vector , the original calculation model of the center displacement of a simply supported beam is , and the proxy model is .

[0133] It should be noted that for the construction of polynomial chaotic expansion of multiple random variables, the expression of multivariate PCE is:

[0134]

[0135]

[0136] Where d is the number of random variables.

[0137] Then in the above-mentioned simply supported beam application scenario, the following polynomial chaos expansion formula is obtained:

[0138]

[0139] in, are multivariate orthogonal polynomials in the sense of normal distribution, which are composed of the corresponding univariate orthogonal polynomials Doing the tensor product yields , in particular, using the normalized Hermite polynomials. is a multivariate indicator. is the set of multivariate indices of the selected orthogonal polynomials. corresponds to the polynomial The expansion coefficient, usually the coefficient It can be given by the projection method. Using the orthogonality of the basis function, it can be calculated:

[0140]

[0141] In the actual application scenario of simply supported beams, according to the preset probability density function , use the preset sampling algorithm to sample and obtain random variables n sample points, and correspondingly calculate their actual observation values, solve the above polynomial chaos expansion formula, and obtain the expansion coefficient The initial value of .

[0142] Afterwards, train the generative model We study the distribution of the absolute value of the inner product of the normalized Galerkin projection separately and find that it changes drastically in a certain area (failure boundary), indicating that it is a high-importance area. Update and resample to obtain each random variable The newly added sample points are concentrated in the area of ​​​​drastic changes, and the newly added sample points are merged with the original sample points to obtain the updated sample data set, and then the above polynomial chaos expansion formula is re-solved to obtain the expansion coefficient The updated value of .

[0143] The iteration cycle is repeated until the iteration termination condition is met, and the final polynomial chaos expansion replacement model of the center displacement (elastic perturbation) of the simply supported beam is obtained. .

[0144] Compared with traditional methods, the method of the present invention is used to study the center displacement problem of a simply supported beam, which effectively reduces the number of simulations, effectively improves the model accuracy, and saves calculation time.

[0145] See also Figure 7 , is a schematic diagram of the structure of a data processing device based on polynomial chaos expansion provided by an embodiment of the present invention. The embodiment of the present invention further provides a data processing device 10 based on polynomial chaos expansion, comprising:

[0146] The random variable determination module 11 is used to determine the original calculation model of the parameters to be solved and the random variables that meet the preset probability density function;

[0147] A sample data set generating module 12 is configured to obtain a plurality of sample points of the random variable according to the probability density function as a sample data set;

[0148] A polynomial chaos expansion module 13 is configured to perform a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and to solve the expansion coefficients in the polynomial chaos expansion.

[0149] A generative model training module 14 is configured to train a plurality of generative models, wherein the generative models are configured to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions;

[0150] The sample data set updating module 15 is configured to update the probability density function according to the generative model, obtain a plurality of new sample points of the random variable according to the updated probability density function, and incorporate the new sample points into the sample data set; and trigger the polynomial chaos expansion module to re-execute the steps of: performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion.

[0151] The proxy model generation module 16 is used to generate a polynomial chaos expansion proxy model of the parameter to be solved according to the expansion coefficient obtained at the last solution when a preset iteration termination condition is met.

[0152] It should be noted that the data processing device based on polynomial chaos expansion provided in an embodiment of the present invention is used to execute all the process steps of the data processing method based on polynomial chaos expansion in the above embodiment. The working principles and beneficial effects of the two correspond one to one, so they will not be repeated here.

[0153] An embodiment of the present invention also provides a data processing device based on polynomial chaotic expansion, comprising a processor, a memory, and a computer program stored in the memory and configured to be executed by the processor. When the processor executes the computer program, it implements the data processing method based on polynomial chaotic expansion as described in any one of the above embodiments.

[0154] An embodiment of the present invention also provides a computer-readable storage medium, which includes a stored computer program, wherein when the computer program is running, the device where the computer-readable storage medium is located is controlled to execute the data processing method based on polynomial chaos expansion as described in any of the above embodiments.

[0155] An embodiment of the present invention also provides a computer program product, which includes a computer program or computer instructions. When the computer program or computer instructions are executed by a processor, the data processing method based on polynomial chaos expansion as described in any of the above embodiments is implemented.

[0156] Those skilled in the art will appreciate that all or part of the processes in the above-described method embodiments can be implemented by instructing the relevant hardware through a computer program. The program can be stored in a computer-readable storage medium, and when executed, the program can include the processes in the above-described method embodiments. The storage medium can be a magnetic disk, an optical disk, a read-only memory (ROM), or a random access memory (RAM).

[0157] The above is a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A data processing method based on polynomial chaos expansion, characterized in that: include: Determine the original calculation model of the parameters to be solved and the random variables that meet the preset probability density function; Obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set; According to the sample data set, the original calculation model is subjected to polynomial chaos expansion using orthogonal polynomial basis functions, and expansion coefficients in the polynomial chaos expansion are solved; Training a plurality of generative models; wherein the generative models are used to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions; updating the probability density function according to the generative model, obtaining a plurality of newly added sample points of the random variable according to the updated probability density function, and incorporating the newly added sample points into the sample data set; Re-execute the steps of: performing a polynomial chaos expansion on the original calculation model according to the sample data set using a preset orthogonal polynomial basis function, and solving the expansion coefficients in the polynomial chaos expansion until a preset iteration termination condition is met, and generating a polynomial chaos expansion proxy model of the parameter to be solved according to the expansion coefficients finally solved; The method of performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set and solving the expansion coefficients in the polynomial chaos expansion includes: Obtaining an actual observation value corresponding to each sample point in the sample data set; According to each of the sample points, a preset orthogonal polynomial basis function is used to perform a polynomial chaotic expansion on the original calculation model to obtain a polynomial chaotic expansion containing the expansion coefficient to be solved; The expansion coefficient is calculated by minimizing the residual sum of squares by the least squares method; wherein the residual sum of squares is obtained by summing the squares of the difference between the polynomial chaotic expansion of each sample point and the corresponding actual observation value; The training of a plurality of generative models, wherein the generative models are used to learn the distribution of the absolute values ​​of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions, includes: Calculating the distribution of absolute values ​​of Galerkin projection inner product terms of the orthogonal polynomial basis functions; Training several generative models to learn the distribution of the normalized absolute value of the Galerkin projection inner product term; The model parameters of the generative model are optimized by minimizing the KL divergence.

2. The data processing method based on polynomial chaos expansion according to claim 1, characterized in that: The step of obtaining a plurality of sample points of the random variable according to the probability density function as a sample data set includes: According to the probability density function, a number of sample points of the random variable are obtained by uniform sampling or Latin hypercube sampling as a sample data set.

3. The data processing method based on polynomial chaos expansion according to claim 1, characterized in that: The preset iteration termination condition is: The number of adaptive iterations reaches the preset iteration threshold; or, The residual sum of squares is less than the preset error threshold.

4. The data processing method based on polynomial chaos expansion according to any one of claims 1 to 3, characterized in that: The parameter to be solved is the center displacement of the simply supported beam; the random variable is at least one of the external force acting on the simply supported beam, the length of the simply supported beam, the elastic modulus and the section moment of inertia.

5. A data processing device based on polynomial chaos expansion, characterized in that: include: A random variable determination module is used to determine the original calculation model of the parameters to be solved and the random variables that meet the preset probability density function; A sample data set generating module, configured to obtain a plurality of sample points of the random variable according to the probability density function as a sample data set; A polynomial chaos expansion module is used to perform polynomial chaos expansion on the original calculation model according to the sample data set using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion; A generative model training module, configured to train a plurality of generative models; wherein the generative models are configured to learn the distribution of the absolute values ​​of the inner product terms of the Galerkin projections of the normalized orthogonal polynomial basis functions; a sample data set updating module, configured to update the probability density function according to the generative model, obtain a plurality of newly added sample points of the random variable according to the updated probability density function, and incorporate the newly added sample points into the sample data set; and trigger the polynomial chaos expansion module to re-execute the steps of: performing a polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion; An agent model generation module is used to generate a polynomial chaos expansion agent model of the parameter to be solved according to the expansion coefficient obtained at the last solution when a preset iteration termination condition is met; The polynomial chaos expansion module is specifically used for: Obtaining an actual observation value corresponding to each sample point in the sample data set; According to each of the sample points, a preset orthogonal polynomial basis function is used to perform a polynomial chaotic expansion on the original calculation model to obtain a polynomial chaotic expansion containing the expansion coefficient to be solved; The expansion coefficient is calculated by minimizing the residual sum of squares by the least squares method; wherein the residual sum of squares is obtained by summing the squares of the difference between the polynomial chaotic expansion of each sample point and the corresponding actual observation value; The generation model training module is specifically used to: Calculating the distribution of absolute values ​​of Galerkin projection inner product terms of the orthogonal polynomial basis functions; Training several generative models to learn the distribution of the normalized absolute value of the Galerkin projection inner product term; The model parameters of the generative model are optimized by minimizing the KL divergence.

6. A data processing device based on polynomial chaos expansion, characterized in that: The method comprises a processor, a memory, and a computer program stored in the memory and configured to be executed by the processor, wherein when the processor executes the computer program, the data processing method based on polynomial chaos expansion as claimed in any one of claims 1 to 4 is implemented.

7. A computer-readable storage medium, characterized in that The computer-readable storage medium includes a stored computer program, wherein when the computer program is executed, the device where the computer-readable storage medium is located is controlled to execute the data processing method based on polynomial chaos expansion according to any one of claims 1 to 4.

8. A computer program product, characterized in that The computer program product includes a computer program or computer instructions, and when the computer program or the computer instructions are executed by a processor, the data processing method based on polynomial chaos expansion according to any one of claims 1 to 4 is implemented.

Images