Method and device for adjusting and optimizing eigenorthogonal decomposition reduced-order modeling sample based on optimal consistent approximation
Through the adaptive sample point optimization method based on the best consistent approximation, the problem of improper selection of sample number and location in the POD method is solved, the model accuracy and computational efficiency are improved, the adaptability to complex environments is enhanced, and more efficient reduced-order modeling is achieved.
Patent Information
- Application Number
- CN202510932087.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-07
- Publication Date
- 2025-10-17
AI Technical Summary
Existing POD methods lack systematic optimization in sample quantity and location selection, resulting in high computational cost, accuracy dependent on sample quality, inability to effectively characterize dynamic or non-stationary systems, and low computational efficiency.
An adaptive sample point optimization method based on best consistent approximation is adopted. The sampling point positions are adjusted through the best consistent approximation algorithm, and the leave-one-out cross-validation method is combined to evaluate the sample quality and optimize the sample point distribution to improve accuracy and robustness.
Without increasing the number of samples, the accuracy and computational efficiency of the POD method are significantly improved, the adaptability to complex and dynamic environments is enhanced, the maximum error is reduced, and the robustness and predictive ability of the model are improved.
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Figure CN120805459A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application belongs to the field of computer aided engineering (CAE) and industrial simulation software, and can be directly applied to engineering multi-physical field fast prediction calculation, and particularly relates to a sample adjustment and optimization method and device for intrinsic orthogonal decomposition reduced-order modeling based on optimal consistent approximation. BACKGROUND
[0002] With the complexity of engineering problems, the increasing requirements for precision and efficiency, and the increasingly significant calculation bottlenecks of traditional numerical methods, model reduction methods have great potential as a key solution. The intrinsic orthogonal decomposition method (POD) is a typical method for reduced-order modeling based on the projection principle, and has very wide applicability. The combination of POD and the finite element method has been used to solve engineering problems in the electromagnetic field for a long time and is currently in a rapid development stage. Related model reduction techniques are mainly used for electromagnetic distribution calculation of transformers and motors, and are also used for problems such as optimization of radio frequency antennas, thermal conduction of semiconductor devices, and construction of magnetic hysteresis models.
[0003] The Institute of Electrical Engineering of the Chinese Academy of Sciences, the University of Lille in France, and the University of Hokkaido in Japan are active in the research of POD theory and application. When selecting snapshots, the above researches all use equidistant sampling methods, and then use a greedy algorithm to select sampling points and calculate the corresponding snapshots. This algorithm does not consider the correlation of snapshots of each sample point, cannot make snapshots fully retain the amount of information, and is not conducive to the application efficiency of the final snapshot set. H. Igarashi's research group applied POD to the analysis of motors and proposed an adaptive snapshot selection method, but this method is still based on equidistant sampling method and has great limitations.
[0004] The existing POD method has the following main problems: first, the calculation accuracy of the POD method is highly dependent on the quality of the samples, including the number and position. The distribution of the samples for reduction needs to reflect the distribution of the actual physical field with the parameters, but before the physical field is calculated, it is impossible to predict how many samples to take and where to sample; second, the generation of snapshots requires singular value decomposition (SVD) of the high-dimensional sample matrix. If the number of samples is too large, the calculation cost will increase significantly, and if the number of samples is too small, the generated snapshots may not be able to fully represent the system, so it is necessary to balance between the number of samples and the calculation cost and accuracy.
[0005] Snapshot generation in traditional POD methods typically relies on empirical or uniform sampling. This approach first sets sampling points at equal intervals within the range of values, typically with a certain number of redundant points. A snapshot is then generated at each sampling point, and a greedy algorithm is used to find the optimal set of points that minimizes error. This results in a POD snapshot set. This approach lacks a systematic optimization strategy, which can lead to sample redundancy or omission of critical dynamic behaviors. Furthermore, for dynamic or non-stationary systems, a static sample set cannot represent time-varying behavior, while frequent sample updates increase computational costs.
[0006] Sample optimization can improve sample representativeness and coverage, reduce the number of samples and computational costs, enhance the snapshot's adaptability to nonlinear systems and dynamic changes, and improve its extrapolation capability and robustness, significantly enhancing the performance of the POD method. Therefore, sample optimization has important theoretical and practical significance and is a key approach to addressing the limitations of the POD method. Summary of the Invention
[0007] To address the above technical issues, the present invention proposes a method and apparatus for sample adjustment and optimization for intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation. This method adaptively searches and optimizes sampling point locations based on the best consistent approximation method, and ultimately utilizes leave-one-out cross-validation to evaluate the samples. This method, performed while essentially estimating the number of snapshots used (also known as the number of modes or orders), minimizes the apparent error while maintaining the same number of snapshots. The proposed method effectively optimizes sample points, significantly improving the accuracy of the POD method and thereby achieving higher modeling and prediction efficiency.
[0008] The technical solutions adopted in the present invention are as follows:
[0009] A sample adjustment optimization method for intrinsic orthogonal decomposition and order reduction modeling based on the best consistent approximation includes: construction of POD order reduction model: first, sampling points are selected at equal intervals, that is, in the interval of each parameter of the data , according to the number of sampling points required , take the sampling interval Generate samples;
[0010] The sampling point positions are adaptively optimized based on the best consistent approximation algorithm.
[0011] A device for adjusting and optimizing samples of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation, comprising:
[0012] Construction module, used to construct the POD reduction model: First, select sampling points at equal intervals, that is, in the intervals of each parameter of the data , according to the number of sampling points required , taking sampling intervals generating samples;
[0013] an optimization module for adaptively optimizing the sampling point positions based on the best uniform approximation algorithm.
[0014] A computing device comprising at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device is caused to perform the best uniform approximation based intrinsic orthogonal decomposition reduced order modeling sample adjustment optimization method.
[0015] A readable storage medium storing program instructions, when the program instructions are read and executed by a computing device, the computing device is caused to perform the best uniform approximation based intrinsic orthogonal decomposition reduced order modeling sample adjustment optimization method.
[0016] Advantages:
[0017] 1) The accuracy of the reduced order model is improved, the overall maximum error is effectively reduced, and the prediction calculation error is suppressed. After the number of POD sampling points is determined, the best uniform approximation method is used for adaptive adjustment of the sampling points, so that the overall maximum error is controllable, and the accuracy of the established POD reduced order model is guaranteed.
[0018] 2) Robustness and computational efficiency are improved. The optimized sample set is based on not changing the number of samples, which avoids the large number of sample points that may be caused by increasing and decreasing sample points. At the same time, the sample optimization method provides better adaptability to local large nonlinear changes, periodic changes and other problems, making it more stable and reliable in complex, dynamic and uncertain environments. BRIEF DESCRIPTION OF DRAWINGS
[0019] Figure 1 A schematic flow chart showing the selection of sample points and snapshot generation in the POD model reduction process;
[0020] Figure 2 A schematic diagram showing the optimization of the POD model reduction sample points using the best uniform approximation polynomial based on the best uniform approximation based intrinsic orthogonal decomposition reduced order modeling sample adjustment optimization method;
[0021] Figure 3 A flow chart showing the optimization of the POD model reduction sample point positions and interpolation polynomials using the best uniform approximation polynomial;
[0022] Figure 4 A schematic diagram showing the optimization of the POD model reduction sample point positions and interpolation polynomials using the best uniform approximation polynomial. DETAILED DESCRIPTION
[0023] Exemplary embodiments of the present disclosure will be described herein below with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it is understood that the present disclosure can be embodied in various forms and should not be limited by the embodiments set forth herein. Rather, these embodiments are provided so that the present disclosure can be thoroughly and completely understood, and so that the scope of the present disclosure can be conveyed to those skilled in the art.
[0024] Currently, the selection of the snapshot subset of POD involves a repeated calculation process of "select sample - construct snapshot - evaluate error", and since the points used at the preset time are often equidistant, in order to improve the accuracy, the number of candidate points is large, resulting in an overall efficiency that is not very ideal. After the selection of points, the conventional POD calculation process no longer adjusts the position of the points, and the snapshot set generated in this way often has a large error. In order to effectively improve the accuracy of POD, the present application proposes a POD reduced-order model optimization method based on optimal uniform approximation.
[0025] Figure 1 A schematic flowchart of sample point selection and snapshot generation in the POD model reduction process is shown. As shown in FIG. 1, the process of sample point selection and snapshot generation includes: Figure 1
[0026] Construction of the POD reduced-order model: first, equidistantly select the sampling points, i.e., determine the parameter interval of the data (such as the time parameter, such as the structure design parameter, such as the excitation parameter, etc.) , according to the number of required sampling points , take the sampling interval to generate samples;
[0027] Then, a greedy algorithm is used to select suitable snapshots.
[0028] The process can also include: in the case where suitable snapshots cannot be selected, adaptively adjusting the number of samples, which can increase the number of sampling points , re-run the above process until the construction of the POD reduced-order model is achieved.
[0029] The greedy algorithm is an algorithm strategy that always chooses the current optimal solution at each step, gradually reaching the global optimal solution through local optimal solutions. When solving a problem, the greedy algorithm always makes the best choice at the moment, without considering the overall optimization. It only makes a local optimal solution in a certain sense. It usually proceeds in a top-down manner, constructing the solution of the problem through a series of local optimal choices. It does not backtrack or consider future possible effects, and cannot obtain the global optimal solution in some problems, but shows high efficiency and simplicity in many practical problems.
[0030] Wherein, the flow of the greedy algorithm is:
[0031] 1) initialization, input parameter space , initial sample set , maximum sample number and error threshold . Then calculate the initial parameter point corresponding to the state snapshot .
[0032] 2) select samples by iteration, the number of iterations is set to the maximum sample number , each iteration selects candidate points from the remaining parameter space , and calculates the projection error for each candidate point:
[0033] ,
[0034] The projection error is used to evaluate the contribution of the candidate point to the current snapshot, and is used as the criterion function for sample point screening, wherein is the current POD mode, is the function value at the position, is the projection error.
[0035] 3) then select the candidate point that minimizes the projection error and update the initial sample set to: , and finally update the state snapshot to: , and recalculate the POD mode .
[0036] 4) iterate until the error is less than the error threshold or the number of iterations reaches .
[0037] 5) after stopping iteration, use the final sample set S corresponding to the snapshot subset to perform singular value decomposition (SVD), extract the first POD mode , and the reduced order model is . Wherein, is the transpose of the first modal matrix, is the full model.
[0038] The SVD step in the above process is:
[0039] Assuming that the system state dimension is , collect snapshots, construct a snapshot matrix by taking each snapshot as a column vector ,Right now ,in It is a snapshot; then Perform singular value decomposition ,in is the left singular vector matrix, and the column vectors are called POD modes, is a rectangular diagonal matrix with singular values on the diagonal , is the right singular vector matrix.
[0040] The above describes the sample point selection and snapshot generation in the POD model reduction process. However, traditional reduced-order modeling generally uses an equidistant sampling method. Determining the number of sampling points determines the sampling position. This method is usually not optimal and there is a lot of room for optimization and adjustment. The present invention introduces the best consistent approximation algorithm to adaptively optimize the sampling point position. Figure 2 The process of sample adjustment optimization for eigenorthogonal decomposition reduced-order modeling based on the best consistent approximation is described.
[0041] The best consistent approximation method process is: given a function , find the polynomial , so that the error satisfies the following relationship:
[0042] .
[0043] in, To solve the interval, is the objective function, It is polynomial.
[0044] The core idea of the best uniform approximation method is to find a function in a given function class that is as uniform as possible close to the target function over the entire approximation interval (the maximum error over the entire interval is minimized), rather than focusing only on the local approximation effect (minimizing the average error or other forms of error metrics).
[0045] When a polynomial is used to interpolate and fit the POD snapshot and calculate the state of any parameter point, the maximum deviation of the interpolation polynomial cannot be controlled. The present invention uses the best consistent approximation polynomial to adjust the selected sampling points, search for new sampling point positions, and adjust the overall process as follows: Figure 2 The detailed process is as shown in Figure 3 As shown, the adjustment diagram is as follows Figure 4 shown.
[0046] The basic idea is to adjust the coefficients of the polynomial in an iterative manner, so that the error value of the maximum point of the error of the polynomial and the target function on the interval gradually decreases until a certain convergence condition is met. The specific steps are as follows:
[0047] 1) Initialize the threshold value and adjustment parameter empirical value of the algorithm, and pre-set the tolerance error , the sample point distance adjustment segment number ;
[0048] 2) Select the initial sample point set , ;
[0049] 3) Construct the reduced-order model based on the initial sample point set ;
[0050] 4) Calculate the error of the reduced-order model in each interval ; where is the error, is the th sample point, indicates the number of loops, i.e. the loop from step 2) to step 5); the interval refers to the interval between two adjacent sample points;
[0051] 5) Find the maximum error , judge , if true, end the sample point set adjustment, is the error of the th interval, is the tolerance error; otherwise, go to the next step;
[0052] 6) Select the sample point to be adjusted according to the error size of the interval on both sides of the sample point, i.e. if adjust the node , otherwise adjust the node ; adjust the distance in the interval and the interval according to the multiples of and (i.e. adjust to , , …, , and , , …, ), and evaluate the error of each interval . If , i.e. no further adjustment is needed, end the sample point set adjustment;
[0053] 7) Adjust the sample point to , is the first cycle is the first sample point is adjusted by the corresponding adjustment amount, that is, a new sample is determined, and then the step 2) is jumped to. After the above steps of adjustment, the overall maximum error is effectively suppressed, so that the final obtained reduced order model is approximated to the ideal level and has the approximate equal ripple characteristic.
[0054] After the reduced order model is constructed, in order to evaluate the sample and snapshot quality, the leave-one-out cross-validation method is used to estimate the interpolation form.
[0055] For the case of sample points, the sample point at the middle position is selected, and the error appearing before and after the sample point is calculated. When the error after removal is smaller, it is believed that the complete use of sample points is prone to overfitting, and then the sample point is removed, and further sample points are taken for further testing.
[0056] In summary, for the intrinsic orthogonal decomposition method, the specific reduced order model obtained by the traditional method is based on the sample point (the sample sampling number and the sampling position are determined) and the corresponding snapshot. The present application proposes a method for optimizing the sample point position based on the best consistent approximation idea, combines the error of the reduced order model, does not change the sample number, effectively improves the precision of the reduced order model by adjusting and optimizing the sampling position, improves the robustness and calculation efficiency of the reduced order modeling, and can especially suppress the maximum error.
[0057] The present application also provides a sample adjustment and optimization device for intrinsic orthogonal decomposition reduced order modeling based on the best consistent approximation, comprising:
[0058] The construction module is used for constructing the POD reduced order model: first, the sample points are selected at equal intervals, that is, the interval of each parameter of the data is determined , the sampling interval is generated according to the number of required sample points to generate samples;
[0059] The optimization module is used for adaptively optimizing the sample point position based on the best consistent approximation algorithm.
[0060] A computing device is also provided, comprising at least one processor and a memory storing program instructions; when the program instructions are read and executed by the processor, the computing device performs the above-mentioned sample adjustment and optimization method for intrinsic orthogonal decomposition reduced order modeling based on the best consistent approximation.
[0061] Also provided is a readable storage medium storing program instructions, which, when read and executed by a computing device, cause the computing device to perform the above-described eigen-orthogonal decomposition reduced-order modeling sample adjustment optimization method based on optimal consistent approximation.
[0062] In the description provided herein, a number of specific details are described. However, it is understood that embodiments of the application can be practiced without these specific details. In some instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.
[0063] Although the application has been described in terms of limited embodiments, it will be apparent that other embodiments, not expressly mentioned herein, can be substituted for those described without departing from the scope of the application as recited in the claims. Moreover, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes and can not have been selected to expressly delineate or otherwise limit the scope of the inventive subject matter.
Claims
1. A sample adjustment optimization method for intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation, characterized in that: include: Construction of POD reduction model: First, select sampling points at equal intervals, that is, determine the parameter intervals of the data , according to the number of sampling points required , take the sampling interval Generate samples; The sampling point positions are adaptively optimized based on the best consistent approximation algorithm.
2. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 1, characterized in that: Adaptive optimization of sampling point locations based on the best consistent approximation algorithm, including: 1) Initialize the algorithm threshold and adjust the parameter experience value, pre-set the tolerance error , the sampling point distance adjusts the number of segments ; 2) Select the initial sampling point set , ; 3) Construct a reduced-order model based on the initial sampling point set ; 4) Calculate the error of the reduced-order model in each interval ;in, is the error, It is Sample points, Indicates the number of cycles, i.e., the cycle from step 2) to step 5); the interval refers to the interval between two adjacent sample points; 5) Find the maximum error ,judge , if it holds true, then the sample point set adjustment ends. It is The error of the interval, Is the tolerance error; otherwise, go to the next step; 6) Select the sample points to be adjusted based on the error between the two sides, that is: Adjust the node , otherwise adjust the node ; Adjust the distance in the interval and interval According to and and evaluate the error of each interval ;when , that is, no further adjustment is required, and the sample point set adjustment is ended; 7) Set the sample points Adjust to , It is Cycle No. The sample points adjusted by the sample points, among which, Indicates the corresponding adjustment amount, that is, determines the new sample , and then skip to step 2).
3. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 1, characterized in that: Also includes: Use a greedy algorithm to select suitable snapshots.
4. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 3, characterized in that: Use a greedy algorithm to select appropriate snapshots, including: Adaptively adjust the number of samples when a suitable snapshot cannot be selected to increase the number of sampling points , repeat the above process until the POD reduction model is constructed.
5. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 3, characterized in that: The process of the greedy algorithm is: 1) Initialize and enter parameter space , initial sample set , maximum number of samples and the error threshold , and then calculate the initial parameter point Corresponding state snapshot ; 2) Select samples through iteration, and the number of iterations is set by To the maximum number of samples , each iteration from the remaining parameter space Select candidate points , and calculate the projection error for each candidate point: ， The projection error is used to evaluate the contribution of candidate points to the current snapshot and serves as a criterion function for sample point screening, where is the current POD mode, For location The function value at is the projection error; 3) Then select the candidate point that minimizes the projection error And update the initial sample set to: , the last updated status snapshot is: , and recalculate the POD modal ; 4) Iterate until the error is less than the error threshold or the number of iterations reaches ; 5) After stopping the iteration, use the snapshot subset corresponding to the final sample set S to perform singular value decomposition SVD to extract the POD mode , the reduced-order model for ;in, It is before The transpose of the modal matrix, It's the full model.
6. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 5, characterized in that: The steps of SVD are: Assume that the system state dimension is ,collection snapshots, taking each snapshot as a column vector and constructing a snapshot matrix ,Right now ,in, It is a snapshot; then Perform singular value decomposition ,in, is the left singular vector matrix, and the column vectors are called POD modes, is a rectangular diagonal matrix with singular values on the diagonal , is the right singular vector matrix.
7. The method for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation according to claim 3, characterized in that: Also includes: After constructing the reduced-order model, the leave-one-out cross-validation method is used to estimate the interpolation shape: In the case of multiple sampling points, the sampling point in the middle is selected and the error before and after removing the sampling point is calculated. When the error after removal is smaller, it is generally considered that the full adoption When there are more than one sampling point, overfitting may occur, and then the sampling point is removed, and then the The sampling points were further inspected.
8. A device for sample adjustment and optimization of intrinsic orthogonal decomposition and reduced-order modeling based on best consistent approximation, characterized in that: include: Construction module, used to construct the POD reduction model: First, select sampling points at equal intervals, that is, in the intervals of each parameter of the data , according to the number of sampling points required , take the sampling interval Generate samples; The optimization module is used to adaptively optimize the sampling point positions based on the best consistent approximation algorithm.
9. A computing device, characterized in that include: at least one processor and memory storing program instructions; When the program instructions are read and executed by the processor, the computing device is caused to execute the sample adjustment optimization method for proper orthogonal decomposition reduced-order modeling based on best consistent approximation according to any one of claims 1 to 7.
10. A readable storage medium storing program instructions, characterized in that: When the program instructions are read and executed by a computing device, the computing device is enabled to perform the sample adjustment optimization method for intrinsic orthogonal decomposition reduced-order modeling based on best consistent approximation according to any one of claims 1 to 7.