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

Abstract

The invention discloses a data processing method and device based on polynomial chaos expansion, a medium and a product, and the method comprises the steps: obtaining a plurality of sample points of a random variable as a sample data set; performing polynomial chaos expansion by adopting an orthogonal polynomial basis function, and solving an expansion coefficient in a polynomial chaos expansion; training a plurality of generative models, and learning distribution of normalized Galerkin projection inner product term absolute values; updating the probability density function according to the generation model, obtaining a plurality of newly added sample points of the random variable according to the updated probability density function, and merging the newly added sample points into the sample data set; and the expansion coefficient in the polynomial chaos expansion is solved again until the iteration termination condition is met, and the polynomial chaos expansion agent model is generated according to the finally solved expansion coefficient. By adopting the method, the fitting precision of the agent model can be improved, the calculation amount is effectively reduced, and the data processing efficiency is improved.

Description

Technical Field

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

[0002] In the fields of engineering and applied sciences, uncertainty quantification (UQ) is a crucial research direction. As an important representative of UQ methods, the polynomial chaos expansion (PCE) technique has been widely applied in various uncertainty quantification problems due to its universality. The core idea of the PCE method is to construct a computationally efficient surrogate model using orthogonal polynomial basis functions to approximately replace a complex original computational model.

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

[0004] However, the inventor found that the prior art has at least the following problems: During the process of using the existing polynomial chaos expansion algorithm to handle practical problems, the sample points of the random variables obtained by least squares sampling will deviate from the true model during calculation, resulting in calculation result errors. Therefore, how to achieve better fitting in the places where the model changes greatly is very crucial, and currently, there is a need to solve similar problems in many important fields such as engineering, finance, and 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 surrogate model, effectively reduce the computational amount, and improve the data processing efficiency.

[0006] To achieve the above purpose, the embodiments of the present invention provide a data processing method based on polynomial chaos expansion, including: Determine the original calculation model of the parameter to be solved and the random variables that satisfy the preset probability density function; Obtain a number of sample points of the random variable according to the probability density function as a sample data set; According to the sample data set, perform polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion formula; Train a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions; Update the probability density function according to the generative model, and obtain a number 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; Re-execute the steps: According to the sample data set, perform polynomial chaos expansion on the original calculation model using the preset orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion formula, until the preset iteration termination condition is satisfied, and generate a polynomial chaos expansion surrogate model of the parameter to be solved according to the finally solved expansion coefficients.

[0007] As an improvement of the above solution, the performing polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions according to the sample data set and solving the expansion coefficients in the polynomial chaos expansion formula includes: Obtain the actual observation value corresponding to each sample point in the sample data set; According to each sample point, perform polynomial chaos expansion on the original calculation model using the preset orthogonal polynomial basis functions to obtain a polynomial chaos expansion formula containing the expansion coefficients to be solved; Calculate the values of the expansion coefficients by minimizing the sum of squared residuals using the least squares method; wherein, the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion formula of each sample point and the corresponding actual observation value.

[0008] As an improvement of the above solution, the training a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions includes: Calculate the distribution of the absolute values of the Galerkin projection inner product terms of the orthogonal polynomial basis functions; Train a number of generative models to learn the distribution of the absolute values of the normalized Galerkin projection inner product terms; Optimize the model parameters of the generative model by minimizing the KL divergence.

[0009] As an improvement of the above solution, obtaining a number 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.

[0010] As an improvement of the above solution, the preset iteration termination condition is: The adaptive iteration times reach the preset iteration times threshold; Or, The sum of squared residuals is less than the preset error threshold, and the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion of each sample point and the corresponding actual observation value.

[0011] As an improvement of the above solution, the parameter to be solved is the central displacement of a simply supported beam; the random variables include the external force acting on the simply supported beam, the length of the simply supported beam, the elastic modulus, and the moment of inertia of the cross section.

[0012] An embodiment of the present invention further provides a data processing device based on polynomial chaos expansion, including: A random variable determination module, configured to determine an original calculation model of a parameter to be solved and a random variable that satisfies a preset probability density function; A sample data set generation module, configured to obtain a number of sample points of the random variable according to the probability density function as a sample data set; A polynomial chaos expansion module, configured to perform polynomial chaos expansion on the original calculation model according to the sample data set by using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion; A generation model training module, configured to train a number of generation models; wherein, the generation model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions; A sample data set update module, configured to update the probability density function according to the generation model, obtain a number of new sample points of the random variable according to the updated probability density function, incorporate the new sample points into the sample data set; and trigger the polynomial chaos expansion module to re-execute the steps: perform polynomial chaos expansion on the original calculation model according to the sample data set by using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion; A surrogate model generation module, configured to generate a polynomial chaos expansion surrogate model of the parameter to be solved according to the finally obtained expansion coefficients when the preset iteration termination condition is satisfied.

[0013] An embodiment of the present invention further provides a data processing device based on polynomial chaos expansion, including 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 chaos expansion as described in any one of the above.

[0014] An embodiment of the present invention further provides a computer-readable storage medium. The computer-readable storage medium includes a stored computer program. When the computer program runs, it controls the device where the computer-readable storage medium is located to execute the data processing method based on polynomial chaos expansion as described in any one of the above.

[0015] An embodiment of the present invention further provides a computer program product. The computer program product includes a computer program or computer instructions. When the computer program or the computer instructions are executed by a processor, they implement the data processing method based on polynomial chaos expansion as described in any one of the above.

[0016] Compared with the prior art, for the data processing method, device, medium, and product based on polynomial chaos expansion disclosed in the present invention, by adopting the technical means of the embodiments of the present invention, a polynomial chaos expansion method based on adaptive importance sampling is used to implement data processing. A generative model is used to learn the absolute value distribution of the Galerkin projection inner product terms after normalization, and an approximate probability density function can be constructed on the random parameter space, and the required samples can be efficiently drawn directly from this generative model. By adaptively generating new sample points iteratively 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 collected samples. While saving costs, due to the reduction in the sample size, the calculation cost of the expansion coefficients is also correspondingly reduced, significantly improving the fitting accuracy and convergence efficiency of the surrogate model. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 is a flowchart of a data processing method based on polynomial chaos expansion provided by an embodiment of the present invention; Figure 2 is a flowchart of a preferred data processing method based on polynomial chaos expansion in an embodiment of the present invention; Figure 3 is a schematic diagram of the principle of uniform sampling; Figure 4 is a schematic diagram of the principle of importance sampling in an embodiment of the present invention; Figure 5 is a schematic diagram of the distribution characteristics of random variables in an embodiment of the present invention; Figure 6 It is a schematic diagram of the distribution of sample sampling points in an embodiment of the present invention; Figure 7 It is a schematic structural diagram of a data processing device based on polynomial chaos expansion provided by an embodiment of the present invention. Detailed implementation manners

[0018] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0019] In the description of the present application, it should be understood that the terms "center", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present application and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus should not be construed as a limitation to the present application.

[0020] The terms "first" and "second" are only used for descriptive purposes, and cannot be understood as indicating or implying relative importance or implicitly specifying the quantity of the indicated technical features. Thus, the features defined with "first" and "second" may explicitly or implicitly include one or more of such features. In the description of the present application, unless otherwise specified, the meaning of "a plurality" is two or more.

[0021] In the description of the present application, it should be noted that unless otherwise clearly defined and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may be a mechanical connection or an electrical connection; it may be directly connected or indirectly connected through an intermediate medium, and it may be the communication inside two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present application can be understood according to specific circumstances.

[0022] See Figure 1 , which is a schematic flowchart 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: S11. Determine the original calculation model of the parameter to be solved and the random variable that satisfies the preset probability density function; S12. Obtain a number of sample points of the random variable according to the probability density function as a sample data set; S13. According to the sample data set, perform polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion formula; S14. Train a number of generation models; wherein, the generation model is used to learn the distribution of the absolute value of the Galerkin projection inner product term of the normalized orthogonal polynomial basis function; S15. Update the probability density function according to the generation model, obtain a number 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; S16. Re-execute the steps: According to the sample data set, perform polynomial chaos expansion on the original calculation model using the preset orthogonal polynomial basis function, and solve the expansion coefficients in the polynomial chaos expansion formula until the preset iteration termination condition is met. Then, generate a polynomial chaos expansion surrogate model for the parameter to be solved according to the finally obtained expansion coefficients.

[0023] It should be noted that the polynomial chaos expansion technology uses orthogonal polynomial basis functions to construct a computationally efficient surrogate model to approximately replace a complex original calculation model. Consider using polynomial chaos expansion to approximate a finite-variance original calculation model . Among them, it is assumed that is a random input variable and it follows a probability density function . At this time, the following polynomial chaos expansion can be used to approximate the original calculation model:

[0024] Among them, is a multivariate orthogonal polynomial selected according to the probability density function or data. These multivariate orthogonal polynomials are obtained by taking the tensor product of the corresponding univariate orthogonal polynomials; is the undetermined expansion coefficient to be solved, usually fitted through existing data and calculated using the least squares method; N is the truncation order, which controls the model complexity. When the data used is sampled strictly according to the probability density function and when , the calculated by the least squares method will tend to the following Galerkin projection, that is:

[0025] However, in many practical problems, the probability density function It is unknown, resulting in deviation from the true model during the sampling process of sample points. For example, when calculating using sample points obtained from the usual uniform distribution or Latin hypercube sampling, errors will occur.

[0026] To solve the above problems, the embodiments of the present invention adopt the method of adaptive importance sampling, and iteratively and adaptively generate sample points with higher quality, so that the areas with larger changes in the model can be better fitted.

[0027] In the embodiments of the present invention, the original calculation model of the parameter to be solved is determined according to the actual problem , and the input value of the original calculation model is used as a random variable, and the random variable follows the distribution characteristics of the probability density function . Since the probability density function is unknown in practical applications, it is first assumed to be a uniform distribution here.

[0028] According to the probability density function , n sample points of the random variable are obtained by sampling using a preset sampling algorithm .

[0029] Preferably, step S12, that is, obtaining a number 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.

[0030] For each of the sample points , a suitable orthogonal polynomial basis function is selected, and the polynomial chaos expansion of the original calculation model is written, that is:

[0031] The expansion coefficients in the polynomial chaos expansion formula are solved . It should be noted that existing means can be used here, such as the least squares method, to solve the expansion coefficients , and a polynomial chaos expansion surrogate model with general accuracy is obtained. The specific means can refer to the existing technology and will not be elaborated here.

[0032] Furthermore, N generative models are used to learn the distribution of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions . Generative models such as Normalizing Flow (NF) can be selected, and the distribution of is approximated by learning the parameter θ, accurately capturing the regions where the function changes violently, and guiding importance sampling.

[0033] According to N generative models Calculate the distribution of the data set and assign it to the probability density function to implement the probability density function for updating. Furthermore, according to the updated probability density function Resample to obtain several new sample points of the random variable. After updating the probability density is higher near the failure boundary, so the new samples will densely appear in this area. These new sample points can better capture the parts with large changes in the function. Add the new sample points to the sample data set and merge them to obtain a new sample data set.

[0034] Then jump to step S13 and re-execute the steps: According to the sample data set, perform polynomial chaos expansion on the original calculation model using a preset orthogonal polynomial basis function, and solve the expansion coefficients in the polynomial chaos expansion formula.

[0035] Iteratively calculate in this way until the preset iteration termination condition is met. According to the finally obtained expansion coefficients Generate a polynomial chaos expansion surrogate model for the parameter to be solved for use in subsequent applications to replace the complex original calculation model to implement the calculation of the parameter to be solved.

[0036] By using the technical means of the embodiment of the present invention, a polynomial chaos expansion method based on adaptive importance sampling is used to implement data processing. A generative model is used to learn the absolute value distribution of the normalized Galerkin projection inner product term, which can construct an approximate probability density function in the random parameter space and efficiently extract the required samples directly from this 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 collected samples. While saving costs, due to the reduction of the sample size, the calculation cost of the expansion coefficients is also correspondingly reduced, significantly improving the fitting accuracy and convergence efficiency of the surrogate model.

[0037] As a preferred implementation manner, 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, performing polynomial chaos expansion on the original calculation model using an orthogonal polynomial basis function and solving the expansion coefficients in the polynomial chaos expansion formula, includes steps S131 to S133: S131. Obtain the actual observed value corresponding to each sample point in the sample data set; S132. For each of the sample points, perform polynomial chaos expansion on the original calculation model using a preset orthogonal polynomial basis function to obtain a polynomial chaos expansion formula containing the expansion coefficients to be solved; S133. Calculate the values of the expansion coefficients by minimizing the sum of squared residuals through the least squares method; wherein, the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion formula for each sample point and the corresponding actual observed value.

[0038] In an embodiment of the present invention, after obtaining a number of sample points by sampling, the actual observed value corresponding to each sample point can be obtained by directly calculating using the original calculation model, or by actual measurement or observation, or based on prior knowledge, etc. , select a suitable orthogonal polynomial basis to write the polynomial chaos expansion formula, and estimate the expansion coefficients by minimizing the following sum of squared residuals through the least squares method:

[0039] If the matrix is denoted as:

[0040] The vector of undetermined coefficients can be obtained as:

[0041] At this time, a polynomial chaos expansion surrogate model with general accuracy is obtained, and the undetermined coefficients can be approximately expressed as:

[0042] In the embodiment of the present invention, the expansion coefficients of the polynomial chaos expansion formula are solved through the least squares method, which can effectively avoid the limitation of sparse grids that need to sample at grid points and can flexibly process high-dimensional data.

[0043] As a preferred embodiment, the embodiment of the present invention is further implemented on the basis of the above embodiment, step S14, that is, training a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis function, including S141 to S143: S141. Calculate the distribution of the absolute values of the Galerkin projection inner product terms of the orthogonal polynomial basis function; S142. Train a number of generative models to learn the distribution of the absolute values of the normalized Galerkin projection inner product terms; S143. Optimize the model parameters of the generative model by minimizing the KL divergence.

[0044] In the embodiments of the present invention, the Galerkin projection inner product term of the orthogonal polynomial basis function is as follows:

[0045] Use generative models to learn the distributions of the absolute values of the normalized Galerkin projection inner product terms respectively, and obtain the parameters in the generative models by minimizing the following KL divergence :

[0046] Since is usually unknown or difficult to sample in high dimensions, use as its operable approximate substitute. The embodiments of the present invention use generative models such as Normalizing Flow (NF) to learn and approximate the distribution of through the parameter θ. NF is a reversible neural network that can transform a simple distribution (such as a Gaussian) into a complex distribution.

[0047] Thus, N generative models are trained for subsequent update of the probability density function and sample collection. The embodiments of the present invention dynamically adjust to approximate the current optimal importance distribution, guiding the next round of sampling to focus more on key regions, avoiding artificially preset distributions (such as Gaussians), and being completely data-driven.

[0048] By adopting the technical means of the embodiments of the present invention, a complex posterior evaluation process is not required. Instead, a generative model is used to learn the absolute value distribution of the normalized Galerkin projection inner product term. Based on this, an approximate probability density function can be constructed on the random parameter space, and the required samples can be efficiently drawn directly from this generative model. By iteratively and adaptively generating new sample points and dynamically integrating them into the overall sample set, and then using the least squares method to solve the undetermined coefficients of the polynomial chaos expansion, this strategy significantly improves the fitting accuracy and convergence efficiency of the surrogate model.

[0049] As a preferred implementation manner, the embodiments of the present invention are further implemented on the basis of any of the above embodiments. The preset iteration termination conditions are as follows: The number of adaptive iterations reaches a preset iteration number threshold; Or, The sum of squared residuals is less than a preset error threshold, and the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion of each sample point and the corresponding actual observed value.

[0050] See Figure 2, which is a schematic flowchart of the preferred data processing method based on polynomial chaos expansion in the embodiments of the present invention. In the embodiments of the present invention, a suitable maximum adaptive iteration number is determined in advance as the iteration number threshold, and / or a minimum sum of squared residuals is determined as the error threshold.

[0051] In the process of iteratively and adaptively generating sample points and calculating expansion coefficients, if the current adaptive iteration number is greater than the preset iteration number threshold, or the sum of squared residuals calculated currently is less than the preset error threshold, it indicates that the iteration termination condition has been satisfied. According to the finally obtained expansion coefficients generate the polynomial chaos expansion surrogate model of the parameter to be solved .

[0052] The following compares the adaptive importance sampling method of the embodiments of the present invention with the traditional uniform sampling method to explain the beneficial effects of the embodiments of the present invention.

[0053] See Figure 3 and Figure 4 , Figure 3 is a schematic diagram of the principle of uniform sampling, Figure 4 is a schematic diagram of the principle of importance sampling in the embodiments of the present invention. First, the used adaptive importance sampling method is described. Suppose an integral needs to be calculated, and the integrand of this integral is a normal distribution with a mean of 0 and a variance of 0.01, and the Monte Carlo method is used to calculate the integral. If the sample points obtained from the uniform distribution are used as the data set, the error of calculating the integral will be very large, as shown in Figure 3 . However, if importance sampling is used and the sample points are sampled from the Gaussian distribution as the data set, the accuracy of the integral 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 parts with larger changes in the integrand, as shown in Figure 3 and Figure 4 . Almost all the samples of importance sampling are distributed in the steep region [-0.5, 0.5].

[0054] In the embodiments of the present invention, the integrand is relatively complex, and the integrand is , and it is not necessarily always positive and may be high-dimensional. In order to calculate more accurate undetermined coefficients, the embodiments of the present invention use a generative model to learn the absolute value of the integrand, and then obtain new sample points from the generative model, so that the newly resampled sample points can better capture the parts with larger changes in each integrand.

[0055] See Figure 5 and Figure 6 ,Figure 5 It is a schematic diagram of the distribution characteristics of random variables in the embodiments of the present invention. Figure 5 It is a schematic diagram of the distribution of sample sampling points in the embodiments of the present invention. Here, the situation of using a generative model to learn the absolute value of a two-dimensional integrand function is shown. It can be observed that the generative model can relatively well capture the function change of the integrand function at the point [0.5, 0.5], as Figure 5 , and at the same time, the new training points obtained by sampling using the generative model are also roughly distributed in the important regions of the function change, as Figure 6 . Therefore, for each individual undetermined coefficient , some sample points that can make the calculation of the Galerkin projection inner product term more accurate can be obtained by training the corresponding generative model. However, these sample points are not directly used to calculate the inner product term, but the sample points that can better capture the characteristics of the true model function are added to the original data set, and the least squares method is used to calculate a new polynomial chaos expansion surrogate model again.

[0056] By adopting the technical means of the embodiments of the present invention, a polynomial chaos expansion method based on adaptive importance sampling is proposed. This method can better capture the characteristics of the true model by reasonably and adaptively selecting sample points, and then calculate the surrogate model more accurately. Each newly generated sample adaptively is added to the original data set and calculated together, with high utilization rate of data and no waste of computing power. By selecting reasonable inner product terms for learning, the statistical quantities of the model, such as mean, variance, etc., can be efficiently calculated through the expansion coefficients.

[0057] The method proposed by the present invention has the following advantages: First, the samples used in the error evaluation process can be recycled, effectively improving the utilization rate of computing resources; second, the adaptive process adopts an incremental sample point addition strategy, completely retaining the historical sample data; finally, the accuracy requirement for the generative model required for sampling is relatively loose, further reducing the computational cost. The present invention has significant advantages in calculating the statistical quantities (such as mean, variance, etc.) of polynomial chaos expansion. By focusing on learning the absolute value probability distribution of the Galerkin projection inner product term corresponding to the target statistical quantity, the calculation accuracy of the statistical quantity is effectively improved.

[0058] As a preferred implementation manner, the embodiments of the present invention are further implemented on the basis of any of the above embodiments. In a specific implementation scenario, the parameter to be solved is the central displacement of a simply supported beam; the random variables are 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 moment of inertia of the cross section.

[0059] In another specific implementation scenario, the parameter to be solved is the cloud feedback parameter, and the random variables include cloud albedo and / or aerosol concentration.

[0060] Understandably, the above scenario is only an optional specific implementation manner. In the actual application process, the data processing method based on polynomial chaos expansion in the embodiments of the present invention can also be applied in other fields. For example, in computational fluid dynamics, it can solve the problem of output uncertainty caused by random parameters (such as viscosity, boundary conditions) in the fluid model, so as to quickly quantify the statistical characteristics of flow field variables (velocity, pressure), and replace expensive direct numerical simulations; or it can be used for climate and earth system modeling to solve the problem that the uncertainty of climate model parameters (such as cloud albedo) affects the prediction results, thereby calibrating the parameter distribution and quantifying the confidence interval of the prediction results; or it can be used for electronic circuit design to solve the problem of circuit performance fluctuations caused by device manufacturing tolerances, and analyze the influence of random parameters on signal integrity and power consumption, etc.

[0061] Exemplarily, taking the structural reliability analysis scenario of a simply supported beam as an example, the embodiments of the present invention provide a data processing method based on polynomial chaos expansion. The method includes steps S21 to S26: S21. Obtain the external force acting on the simply supported beam, the length, elastic modulus, and cross-sectional moment of inertia of the simply supported beam as random variables, and take the central displacement of the simply supported beam as the parameter to be solved, and obtain the original calculation model of the central displacement of the simply supported beam; wherein, the random variables satisfy a preset probability density function. S22. Obtain a number of sample points of the random variables according to the probability density function as a sample data set. S23. Perform polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula. S24. Train a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions. S25. Update the probability density function according to the generative model, and obtain a number of new sample points of the random variables according to the updated probability density function, and incorporate the new sample points into the sample data set. S26. Re-execute the steps: perform polynomial chaos expansion on the original calculation model using preset orthogonal polynomial basis functions according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula until a preset iteration termination condition is met, and generate a polynomial chaos expansion surrogate model of the central displacement of the simply supported beam according to the finally solved expansion coefficients.

[0062] Preferably, based on the sample data set, performing polynomial chaos expansion on the original calculation model by using orthogonal polynomial basis functions, and solving the expansion coefficients in the polynomial chaos expansion formula, includes: Obtaining the actual observed value corresponding to each sample point in the sample data set; Performing polynomial chaos expansion on the original calculation model by using the preset orthogonal polynomial basis functions according to each sample point, to obtain a polynomial chaos expansion formula including the expansion coefficients to be solved; Calculating the values of the expansion coefficients by minimizing the sum of squared residuals through the least squares method; wherein, the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion formula of each sample point and the corresponding actual observed value.

[0063] Preferably, training a plurality of generation models; wherein, the generation 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 the absolute values of the Galerkin projection inner product terms of the orthogonal polynomial basis functions; Training a plurality of generation models to learn the distribution of the absolute values of the normalized Galerkin projection inner product terms; Optimizing the model parameters of the generation models by minimizing the KL divergence.

[0064] Preferably, the preset iteration termination condition is: the adaptive iteration number reaches a preset iteration number threshold; or, the sum of squared residuals is less than a preset error threshold.

[0065] 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, assume is the external force acting on the beam, is the length of the beam, is the elastic modulus of the material, is the moment of inertia of the beam cross-section, then the central displacement (elastic deflection) of the beam is:

[0066] Further assume that the external force satisfies a normal distribution with a mean of , and a standard deviation of , the length of the beam satisfies a normal distribution with a mean of , and a standard deviation of , the elastic modulus satisfies a normal distribution with a mean of , and a standard deviation of 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 .

[0067] It should be noted that for the construction of polynomial chaotic expansion of multiple random variables, the expression of multivariate PCE is:

[0068]

[0069] Where d is the number of random variables.

[0070] Then in the above-mentioned simply supported beam application scenario, the following polynomial chaos expansion formula is obtained:

[0071] 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:

[0072] 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 .

[0073] 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 region with drastic changes. The newly added sample points are merged with the original sample points to obtain an updated sample data set, and then the above polynomial chaos expansion formula is solved again to obtain the expansion coefficients. The updated values.

[0074] Iterate in this way until the iteration termination condition is met, and obtain the polynomial chaos expansion surrogate model of the central displacement (elastic deflection) of the simply supported beam finally. 。

[0075] Using the method of the present invention to study the central displacement problem of a simply supported beam, compared with the traditional method, the number of simulations is effectively reduced, the model accuracy is effectively improved, and the calculation time is saved.

[0076] See Figure 7 , which is a schematic structural diagram of a data processing device based on polynomial chaos expansion provided by an embodiment of the present invention. An embodiment of the present invention also provides a data processing device 10 based on polynomial chaos expansion, including: A random variable determination module 11, configured to determine the original calculation model of the parameter to be solved and the random variables that satisfy the preset probability density function; A sample data set generation module 12, 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 13, configured to perform polynomial chaos expansion on the original calculation model according to the sample data set by using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion formula; A generation model training module 14, configured to train a plurality of generation models; wherein, the generation model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions; A sample data set update module 15, configured to update the probability density function according to the generation 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: perform polynomial chaos expansion on the original calculation model according to the sample data set by using orthogonal polynomial basis functions, and solve the expansion coefficients in the polynomial chaos expansion formula; A surrogate model generation module 16, configured to generate a polynomial chaos expansion surrogate model of the parameter to be solved according to the finally obtained expansion coefficients when the preset iteration termination condition is met.

[0077] It should be noted that a data processing device based on polynomial chaos expansion provided by an embodiment of the present invention is used to execute all the process steps of a data processing method based on polynomial chaos expansion in the above embodiment. The working principles and beneficial effects of the two correspond one by one, so they will not be elaborated here.

[0078] An embodiment of the present invention further provides a data processing device based on polynomial chaos expansion, including 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 chaos expansion described in any one of the above embodiments.

[0079] An embodiment of the present invention further provides a computer-readable storage medium. The computer-readable storage medium includes a stored computer program. When the computer program runs, it controls the device where the computer-readable storage medium is located to execute the data processing method based on polynomial chaos expansion described in any one of the above embodiments.

[0080] An embodiment of the present invention further provides a computer program product. The computer program product includes a computer program or computer instructions. When the computer program or the computer instructions are executed by a processor, they implement the data processing method based on polynomial chaos expansion described in any one of the above embodiments.

[0081] Those of ordinary skill in the art can understand that all or part of the processes in implementing the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The program can be stored in a computer-readable storage medium. When the program is executed, it can include the processes of the embodiments of the above methods. Among them, the storage medium can be a magnetic disk, an optical disk, a read-only memory (ROM), or a random access memory (RAM), etc.

[0082] The above is the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art of this technology, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements are also regarded as the protection scope of the present invention.

Claims

1. A data processing method based on polynomial chaos expansion, characterized in that, Including: Determine the original calculation model of the parameter to be solved and the random variables that satisfy the preset probability density function; Obtain a number of sample points of the random variable according to the probability density function as a sample data set; Perform polynomial chaos expansion on the original calculation model using the orthogonal polynomial basis function according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula; Train a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis function; Update the probability density function according to the generative model, and obtain a number 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; Re-execute the steps: perform polynomial chaos expansion on the original calculation model using the preset orthogonal polynomial basis function according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula, until when the preset iteration termination condition is met, generate a polynomial chaos expansion surrogate model of the parameter to be solved according to the finally solved expansion coefficients.

2. The data processing method based on polynomial chaos expansion according to claim 1, characterized in that The performing polynomial chaos expansion on the original calculation model using the orthogonal polynomial basis function according to the sample data set, and solving the expansion coefficients in the polynomial chaos expansion formula includes: Obtain the actual observed value corresponding to each sample point in the sample data set; Perform polynomial chaos expansion on the original calculation model using the preset orthogonal polynomial basis function according to each sample point to obtain a polynomial chaos expansion formula containing the expansion coefficients to be solved; Calculate the values of the expansion coefficients by minimizing the sum of squared residuals through the least squares method; wherein, the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion formula of each sample point and the corresponding actual observed value.

3. The data processing method based on polynomial chaos expansion according to claim 1, wherein, The training a number of generative models; wherein, the generative model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis function includes: Calculate the distribution of the absolute values of the Galerkin projection inner product terms of the orthogonal polynomial basis function; Train a number of generative models to learn the distribution of the absolute values of the normalized Galerkin projection inner product terms; Optimize the model parameters of the generative model by minimizing the KL divergence.

4. The data processing method based on polynomial chaos expansion according to claim 1, wherein The obtaining a number 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, obtain a number of sample points of the random variable by uniform sampling or Latin hypercube sampling as a sample data set.

5. The data processing method based on polynomial chaos expansion according to claim 1, characterized in that, The preset iteration termination condition is: The adaptive iteration number reaches the preset iteration number threshold; Or, The sum of squared residuals is less than the preset error threshold, and the sum of squared residuals is obtained by summing the squares of the differences between the polynomial chaos expansion formula of each sample point and the corresponding actual observed value.

6. The data processing method based on polynomial chaos expansion according to any one of claims 1 to 5, characterized in that The parameter to be solved is the central displacement of a simply supported beam; the random variables are 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 moment of inertia of the cross section.

7. A data processing device based on polynomial chaos expansion, characterized in that, It includes: A random variable determination module, configured to determine the original calculation model of the parameter to be solved and the random variables that satisfy a preset probability density function; A sample data set generation module, configured to obtain a number of sample points of the random variables according to the probability density function as a sample data set; A polynomial chaos expansion module, configured to perform polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula; A generation model training module, configured to train a number of generation models; wherein, the generation model is used to learn the distribution of the absolute values of the Galerkin projection inner product terms of the normalized orthogonal polynomial basis functions; A sample data set update module, configured to update the probability density function according to the generation model, obtain a number of new sample points of the random variables 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: perform polynomial chaos expansion on the original calculation model using orthogonal polynomial basis functions according to the sample data set, and solve the expansion coefficients in the polynomial chaos expansion formula; A surrogate model generation module, configured to generate a polynomial chaos expansion surrogate model of the parameter to be solved according to the finally solved expansion coefficients when a preset iteration termination condition is satisfied.

8. A data processing device based on polynomial chaos expansion, characterized in that, It includes 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 chaos expansion as described in any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that The computer-readable storage medium includes a stored computer program, wherein when the computer program runs, it controls the device where the computer-readable storage medium is located to execute the data processing method based on polynomial chaos expansion as described in any one of claims 1 to 6.

10. 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, they implement the data processing method based on polynomial chaos expansion as described in any one of claims 1 to 6.

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