Engineering index oriented parameter sensitivity driven reduced order modeling and spatio-temporal attribution method

Abstract

The application discloses an engineering index-oriented parameter sensitivity driven reduced-order modeling and space-time attribution method, and belongs to the technical field of computational fluid dynamics and data-driven modeling. The main steps are as follows: firstly, the original data is preprocessed, then the POD decomposition based on snapshots is executed to build a data set, a full connection residual neural network is trained and its gradient characteristics are evaluated, the parameter sensitivity is sorted through the identification of the active subspace, and then the quantitative diagnosis of the space-time sensitivity field is carried out, finally, the PRBF proxy model is constructed, and the QoI reconstruction can be completed based on the PRBF model. The application constructs a set of flow field reduced-order modeling and diagnosis methods which are oriented to engineering indexes, have extremely low dimensions and have an interpretable physical attribution ability through the active subspace guided POD modal truncation, the construction of a low-dimensional high-precision PRBF proxy model, efficient automatic differentiation gradient calculation and space-time sensitivity field back projection, and has a wide application prospect.

Description

Technical Field

[0001] This invention belongs to the field of computational fluid dynamics and data-driven modeling technology, specifically involving a parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method oriented towards engineering indicators. Background Technology

[0002] With the increasing demands for refined design in industries such as aerospace and energy, numerical simulation of complex flow phenomena has gradually become a core tool in engineering research and development. In modern industrial processes, tasks such as engineering design optimization, uncertainty quantification, digital twins, and real-time control often require parametric scanning of thousands of operating conditions. This large-scale, high-frequency demand for numerical simulation poses a severe challenge to the efficiency and cost of traditional high-fidelity computational methods. Although full-order models (FOMs) can provide extremely high simulation accuracy, their computational cost is extremely high in the aforementioned large-scale repetitive solution scenarios, severely restricting the speed of design iteration. Therefore, although full-order models (FOMs) can provide extremely high simulation accuracy, their computational cost is extremely high in the aforementioned large-scale repetitive solution scenarios, severely restricting the efficiency of design iteration. To alleviate the contradiction between computational accuracy and computational efficiency, data-driven reduced-order models (ROMs) have emerged. The core idea lies in constructing a mathematical model with a computational scale much smaller than FOM (Form of Memory) while still capturing the main physical characteristics by mining the low-dimensional coherent structure contained in high-dimensional flow field data. In the field of fluid mechanics, Proper Orthogonal Decomposition (POD) is currently the most complete and widely used linear order reduction method. Moreover, POD can extract the energy-optimal orthogonal basis through eigenvalue decomposition or singular value decomposition (SVD), achieving efficient compression and reconstruction of large-scale unsteady flow fields. In the past few decades, researchers have greatly promoted the progress of order reduction analysis in fluid engineering based on various ROM techniques of POD.

[0003] However, although POD performs excellently in flow field reconstruction, existing energy-based order reduction frameworks reveal the following three fundamental shortcomings in practical applications aimed at engineering performance analysis:

[0004] 1) There is a fundamental disconnect between energy ranking and engineering metric sensitivity ranking: Traditional POD methods use modal energy (eigenvalues) as the sole criterion for ranking and truncation, implicitly assuming that the highest-energy mode contributes the most to the Quantity of Interest (QoI). However, modal energy essentially reflects the statistical intensity of the kinematic characteristics of the flow field, while engineering QoI (e.g., total pressure loss coefficient, lift coefficient, heat transfer, etc.) is often determined by specific physical mechanisms in the flow field. Numerous experiments and numerical simulations have demonstrated that the two are not equivalent: some modes carrying lower energy may have a much greater sensitivity contribution to QoI than the energy-dominant mode. If a reduced-order model is constructed based on energy truncation, modal information redundancy will inevitably occur, leading to a "performance paradox"—that is, hundreds of modes are retained, and extremely high energy coverage is achieved, but the engineering QoI cannot be reconstructed efficiently and accurately. For example, the patent (application number 202311473688.0) discloses a "fluid-structure interaction method for propellers based on a POD-reduced model." While its goal is to improve the computational accuracy and efficiency of propeller aerodynamics, the construction process of its reduced-order model strictly follows the traditional energy share criterion (selecting modes with an energy share of over 99%). This method aims to predict the key engineering indicator of "propeller aerodynamics," but its mode selection mechanism relies solely on the overall energy distribution of the flow field, rather than the sensitivity of the specific QoI of "aerodynamics." This may lead to a potential problem: some flow modes that contribute significantly to the formation of aerodynamics but have a low spatial energy share (e.g., local small-scale separated vortices) are discarded during the reduction process. This results in the constructed reduced-order model performing well in flow field reconstruction but failing to guarantee efficient and accurate prediction of the engineering indicator of aerodynamics, which is a typical manifestation of the aforementioned "performance paradox."

[0005] 2) There is a sharp contradiction between model dimensionality compression efficiency and subsequent computational overhead: In real turbulent flow fields, energy distribution between modes is extremely diffuse (following Kolmogorov's energy spectrum characteristics). To ensure the accuracy of flow field reconstruction, traditional energy criteria often need to retain hundreds or even thousands of modes. This means that the surrogate model constructed subsequently must be trained in an extremely high-dimensional input space. High-dimensional space not only brings severe "curse of dimensionality," significantly increasing the demand for expensive training samples, but also leads to a sharp increase in computational overhead during online deployment, making it difficult to meet the efficiency requirements of real-time control or large-scale multidisciplinary optimization.

[0006] 3) Existing order reduction frameworks lack physical interpretability and spatiotemporal attribution capabilities: Current order reduction models are mostly treated as "black box" or "semi-black box" processes, only providing the contribution of each mode to the global energy. For engineers, it is impossible to quantitatively project the impact of specific modes on QoI back to the physical spatiotemporal domain, thus making it difficult to identify "when, where, and which local flow structures have the most significant impact on engineering performance." This lack of diagnostic capability makes it difficult for order reduction models to truly guide physical design improvements and to provide deep physical insights into the formation mechanisms of engineering indicators.

[0007] In summary, existing energy-based POD reduction frameworks have significant theoretical and application shortcomings in flow field analysis and design scenarios oriented towards engineering indicators. How to construct a flow field representation method that is sensitivity-guided, has object attribution capabilities, and is extremely low-dimensional while maintaining engineering practicality is a key technical challenge that urgently needs to be solved in the field of fluid mechanics reduction modeling. Summary of the Invention

[0008] The purpose of this invention is to solve the above problems and provide a parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators.

[0009] The technical solution of this invention is: a parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators, comprising the following steps:

[0010] (a) Preprocess the raw data and then perform snapshot-based POD decomposition to construct the dataset.

[0011] (ii) Train a fully connected residual neural network and evaluate its gradient properties.

[0012] (iii) By identifying the active subspace, the sensitivity of parameters can be ranked.

[0013] (iv) Quantitative diagnosis of spatiotemporal sensitive fields.

[0014] (v) After constructing the PRBF proxy model, QoI (attention volume) reconstruction is completed based on the PRBF model.

[0015] Furthermore, in the above-mentioned parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators, the specific operation process of step (I) is as follows: 1) For high-fidelity unsteady flow field simulation data, construct a snapshot matrix.

[0016]

[0017] As the basic input for subsequent eigenorthogonal decomposition, where n t For the number of time snapshots,

[0018]

[0019] This represents the total degrees of freedom.

[0020] 2) For pulsating fields

[0021]

[0022] Apply a positive definite weight matrix with the grid cell area as the diagonal element.

[0023]

[0024] Define weighted inner product

[0025]

[0026] Perform weighted snapshot POD decomposition.

[0027] 3) Determine the optimal time eigenfunction ψ by solving the Fredholm integral equation. n (t), retaining the first n based on energy cutoff. m One dominant mode, obtain spatial mode φ n (ξ), time coefficient a n (t) and singular value σ n It satisfies the orthogonal normalization condition:

[0028]

[0029] Furthermore, in the above-mentioned parameter sensitivity-driven order reduction modeling and spatiotemporal attribution method for engineering indicators, the specific operation process of step (ii) is as follows: 1) Using n m Dimensional POD time coefficient vector

[0030]

[0031] Using the input as input, a fully connected residual neural network is constructed after randomly partitioning the dataset, as shown in the following formula:

[0032]

[0033] in, For the input embedding layer, N1,…,N L Let P be a residual block and P be the output projection layer; each residual block satisfies

[0034]

[0035] It is an identity operator;

[0036] 2) Using the mean squared error, the formula is as follows:

[0037]

[0038] The loss function is used, and the Adam optimizer is used for training. Batch normalization (BN) and early stopping mechanism are introduced to prevent overfitting.

[0039] 3) After training converges, the gradient vector is efficiently calculated using inverse mode automatic differentiation (AD), as shown in the following formula:

[0040]

[0041] The accuracy of the above operations can reach machine accuracy, and the computational efficiency can be improved by about 465 times compared with the finite difference (FD) method.

[0042] Specifically, in step (ii), the dataset is randomly divided into 80% test set, 10% validation set, and 10% test set.

[0043] Furthermore, in the above-mentioned parameter sensitivity-driven order reduction modeling and spatiotemporal attribution method for engineering indicators, the specific operation process of step (iii) is as follows: 1) Construct an estimate of n from gradient samples. m ×n m The gradient covariance matrix is ​​given by the following formula: 2) Perform eigenvalue decomposition on matrix C, as shown in the following formula: 3) Extract the first n feature vectors Zhang Cheng's activity subspace, with 4) Finally, the activity scores for each POD modality are normalized and calculated using the following formula: Specifically, sensitivity-driven mode ranking is achieved by comprehensively quantifying the sensitivity contribution of the i-th POD mode in all activity directions.

[0044] Furthermore, in the above-mentioned parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators, the specific operation process of step (iv) is as follows: 1) Through the active subspace projection operator

[0045]

[0046] The gradient is projected onto the physical domain to generate a spacetime sensitive field;

[0047]

[0048] in:

[0049]

[0050] For the projected gradient vector, and

[0051]

[0052] Let it be its k-th component;

[0053] 2) Statistically analyze the distribution of sensitivity extreme points of all snapshots, and then quantitatively identify the key spatial sites for loss generation and suppression.

[0054] Furthermore, in the above-mentioned parameter sensitivity-driven order reduction modeling and spatiotemporal attribution method for engineering indicators, the specific operation process of step (v) is as follows: 1) convert n t Projecting each training sample onto an n-dimensional activity subspace yields a set of activity variables:

[0055]

[0056] After randomly partitioning the dataset, a multinomial enhanced radial basis function (PRBF) surrogate model is constructed on this low-dimensional space:

[0057]

[0058] 2) The kernel function takes the anisotropic squared exponential form:

[0059]

[0060] Length scale Determined by maximum likelihood estimation, it is inversely proportional to the j-th eigenvalue of the active subspace; polynomial basis For monomials whose total degree does not exceed d;

[0061] 3) Apply orthogonal constraints, as shown in the following formula:

[0062]

[0063] To ensure well-posedness, the following block linear system is formed:

[0064]

[0065] In this case, a small regularization term is added to the diagonal of the kernel matrix K. The weights are then obtained by solving the problem. and ;

[0066] 4) Passing new input Direct evaluation enables rapid QoI prediction.

[0067] Specifically, in step (v), the dataset is randomly divided into 80% training set and 20% test set.

[0068] Compared with the prior art, the technical solution of the present invention has the following significant beneficial effects and advantages:

[0069] 1) This invention introduces the active subspace method into the flow field POD reduced-order modeling framework, realizing sensitivity-guided modal truncation. Specifically, it identifies a significant difference between the sensitivity-dominant mode and the energy-dominant mode, thereby overcoming the inherent limitations of traditional energy truncation methods in QoI reconstruction and more directly reflecting the physical control mechanism of aerodynamic losses.

[0070] 2) This invention constructs a PRBF surrogate model on an extremely low-dimensional active subspace, achieving a significant dimensionality compression compared to the high-dimensional space required by traditional energy cutoff methods. At the same time, the model achieves extremely high fitting accuracy on the test set, which is significantly better than the polynomial response surface under the same dimension, thus taking into account the dual advantages of high accuracy and low-dimensional representation.

[0071] 3) This invention uses FC-ResNet combined with inverse mode automatic differentiation to calculate gradients. Compared with the finite difference method, it can significantly improve computational efficiency while maintaining machine accuracy, providing an efficient means for gradient evaluation of large-scale flow field samples.

[0072] 4) This invention projects modal sensitivity information onto the physical domain through spatiotemporal sensitivity field diagnosis, quantitatively identifying key spatial locations in the flow field that have the most significant impact on engineering performance. Based on this, loss amplification and loss suppression mechanisms are synchronously triggered by the same unsteady flow events, thus providing an interpretable physical basis for engineering design optimization.

[0073] 5) The framework of this invention has good versatility. It can be extended to various turbomachinery flow configurations and is applicable to other complex turbulent flow conditions. It also supports sensitivity analysis and proxy modeling of different engineering QoI, thus having broad application prospects. Attached Figure Description

[0074] Figure 1 This is a flowchart of the method of the present invention;

[0075] Figure 2 This is a schematic diagram of the fully connected residual block (FC Residual Block) structure of the present invention;

[0076] Figure 3 This is a flowchart illustrating the construction process of the PRBF proxy model of the present invention.

[0077] Figure 4 This is a ranking chart of normalized activity scores for each POD mode in this invention;

[0078] Figure 5 The spatiotemporal sensitivity field S(x,y,t) at a typical moment in this invention ᵢ ) Schematic diagram;

[0079] Figure 6 This is a schematic diagram of the high-fidelity QoI reconstruction result of the low-dimensional manifold of active variables composed of sensitivity-dominated POD modes in the algorithm framework of this invention. Detailed Implementation

[0080] To enable those skilled in the art to better understand the technical solutions of the present invention, the technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and specific examples. It should be noted that the embodiments described herein are merely some embodiments of the present invention, and not all embodiments; based on the embodiments disclosed in the present invention, those skilled in the art can obtain various other similar examples and specific solutions without creative effort.

[0081] like Figures 1-6 As shown, to verify the performance of the parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators proposed in this invention, the data flow of five modules—snapshot POD, FC-ResNet training and gradient evaluation, active subspace analysis, spatiotemporal sensitivity diagnosis, and PRBF surrogate model construction—is demonstrated. A three-dimensional compressible T106A low-pressure turbine blade is used as the verification object (this invention is also applicable to other flow configurations such as compressors, airfoils, and duct flow). The specific method is as follows:

[0082] 1) Calculation configuration (preprocessing the raw data):

[0083] Export Reynolds Number:

[0084]

[0085] Export Mach number:

[0086]

[0087] Total inlet pressure:

[0088]

[0089] Outlet static pressure:

[0090]

[0091] Import total temperature:

[0092]

[0093] Axial chord length:

[0094]

[0095] Aspect ratio:

[0096]

[0097] High-fidelity implicit large eddy simulation (ILES) was performed using the spectral element method of polynomial order p3, with a total degree of freedom of 6,532,608.

[0098] 2) Perform snapshot-based POD decomposition to construct a dataset: spanning the near-wake region across the cross-section from the middle of the blade.

[0099]

[0100] Extract n t =10000 time-resolved snapshots, sampling time step

[0101]

[0102] A snapshot is taken every 120 computational steps, with the Mach number field at each instant representing a snapshot. Snapshot matrix.

[0103]

[0104] Weight matrix W o Take the area of ​​each unit. Determine n based on energy cutoff after decomposing the snapshot POD. m Obtain the POD coefficient vector

[0105]

[0106] QoI is defined as the area-average total pressure loss coefficient.

[0107]

[0108] Integrating the cumulative dissipation effects of boundary layer development, wake mixing, secondary flow, and vortex structure into a single scalar is the primary optimization objective for low-pressure turbine aerodynamic design.

[0109] 3) Train a fully connected residual neural network (FC-ResNet) and evaluate its gradient properties: The network consists of an input embedding layer, 7 residual blocks, each containing 2 fully connected layers, and hidden layer dimensions:

[0110]

[0111] The diagram includes Batch Normalization (BN), ReLU, and an output projection layer. The mean squared error is used as the loss function. The Adam optimizer is trained with a learning rate of 1×10⁻³. The early stopping tolerance parameter is:

[0112]

[0113] patience = 50 rounds. Gradient

[0114]

[0115] Automatic differentiation calculation using PyTorch's inverse mode took 40.7 seconds for 10,000 samples and 18,912.0 seconds for finite difference control, resulting in a speedup of approximately 465 times.

[0116] 4) By identifying the active subspace, the parameter sensitivity is ranked, and the covariance matrix is ​​estimated from 10,000 gradient samples.

[0117]

[0118] The eigenvalue spectrum exhibits spectral gaps with significant orders of magnitude spans between the first few orders. The dimension of the active subspace is determined through grid search.

[0119]

[0120] Activity score analysis showed that Mode2 and Mode1 had the highest sensitivity, followed by Mode10 and Mode9; there was a significant deviation between the sensitivity ranking and the energy ranking, which verified the necessity of sensitivity-guided truncation.

[0121] 5) Quantitative diagnosis of spatiotemporal sensitivity field: Calculation of spatiotemporal sensitivity field for 10,000 time points:

[0122]

[0123] Sensitivity positive extreme value:

[0124]

[0125] The snapshots consistently show a concentration near the trailing edge of the blade, confirming that the trailing edge region is the primary spatial location for loss generation and suppression.

[0126]

[0127] It exhibits near-antisymmetric synchronous fluctuations, revealing a bidirectional sensitivity response mechanism of trailing vortex shedding to total pressure loss.

[0128] 6) After constructing the PRBF surrogate model, QoI reconstruction was performed based on this PRBF model: a first-order polynomial enhanced PRBF (p=1) was constructed on the 18-dimensional activity subspace, and the length scale was determined by maximum likelihood estimation optimization. The performance comparison between the PRBF surrogate model and the control method is shown in Table 1:

[0129]

[0130] As shown in Table 1, the PRBF proxy model (N) of this invention... AS (p=18, p=1) Achieved the following on both the training and test sets:

[0131]

[0132] Compared to polynomial response surfaces:

[0133]

[0134] The accuracy was significantly improved, verifying the RBF core's efficient ability to capture nonlinear QoI.

[0135] Table 2 compares the gradient calculation methods of the reverse mode automatic differentiation and the finite difference method.

[0136]

[0137] As shown in Table 2, the reverse mode automatic differentiation method of the present invention significantly reduces the computation time from 18912 seconds to 40.7 seconds, improving the computation efficiency by approximately 465 times. While maintaining the accuracy of gradient calculation, it achieves an order-of-magnitude acceleration in computation, greatly reducing time costs.

[0138] Table 3 compares the sensitivity-guided cutoff method with the traditional energy cutoff method.

[0139]

[0140] As shown in Table 3, this invention requires only 18 modes, far fewer than the 831 modes required by traditional methods, reducing the number of modes by approximately 97.8%, which significantly reduces the complexity and computational scale of the model. Traditional methods use "cumulative energy ≥ 99%" as an indicator, while this invention uses R² > 0.999 as the accuracy standard for the surrogate model, achieving an ultra-high fitting degree of 0.9996. While significantly reducing the number of modes, it still maintains an extremely high level of accuracy.

[0141] In summary, this invention achieves an acceleration of approximately 465 times in gradient calculation efficiency; and in modal truncation and compression, it achieves higher accuracy (R²=0.9996) with a small number of dominant modes. Overall, through the dual optimization of "dimensionality reduction + acceleration," it achieves an order-of-magnitude improvement in computational efficiency while maintaining or even improving accuracy, demonstrating an overwhelming technical advantage over traditional methods.

[0142] Thus, by employing the technical solution of this invention, the active subspace method is introduced into the flow field POD reduced-order modeling framework for the first time, achieving sensitivity-guided modal truncation. The identified sensitivity-dominant modes differ significantly from the energy-dominant modes, overcoming the inherent limitations of traditional energy truncation methods in QoI reconstruction and more directly reflecting the physical control mechanism of aerodynamic losses. Based on this, this invention constructs a PRBF surrogate model on an 18-dimensional active subspace, achieving approximately 46 times dimensionality compression compared to the 831 dimensions (99% energy threshold) required by traditional energy truncation methods. Simultaneously, it achieves R²=0.9996 on the test set, significantly outperforming the same-dimensional polynomial response surface (R²=0.9255), balancing high accuracy with extremely low dimensionality. Furthermore, this invention utilizes FC-ResNet combined with inverse mode automatic differentiation to calculate gradients, with a total processing time of only 40.7 seconds, approximately 465 times faster than the finite difference method (18912 seconds) while maintaining machine accuracy, providing an efficient means for gradient evaluation of large-scale flow field samples. Finally, this invention projects the modal sensitivity information system back to the physical domain through spatiotemporal sensitivity field diagnosis, quantitatively identifies the key spatial locations in the flow field that have the most significant impact on engineering indicators, and discovers that the loss amplification and loss suppression mechanisms are synchronously triggered by the same unsteady flow events, thus providing an interpretable physical basis for engineering design optimization.

[0143] As can be seen from the above description, compared with the prior art, the framework of the present invention is universal and can be extended to various turbomachinery flow configurations or other complex turbulent flow conditions, as well as sensitivity analysis and proxy modeling of different engineering QoI, with broad application prospects.

[0144] The technical solution, working process, and implementation effects of the present invention have been described in detail above. It should be noted that the described examples are only typical examples of the present invention. In addition, the present invention may have many other specific implementation methods. All technical solutions formed by equivalent substitution or equivalent transformation fall within the scope of protection claimed by the present invention.

Claims

1. A parameter-sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators, characterized in that, Includes the following steps: S1: Preprocess the raw data, then perform snapshot-based POD decomposition to build the dataset; S2: Train a fully connected residual neural network and evaluate its gradient properties; S3: By identifying the active subspace, the parameter sensitivity is ranked. S4: Quantitative diagnosis of spatiotemporal sensitivity fields; S5: After constructing the PRBF proxy model, complete the QoI reconstruction based on the PRBF model.

2. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 1, characterized in that, S1 includes the following steps: 1) Construct a snapshot matrix for high-fidelity unsteady flow field simulation data. , which serves as the basic input for subsequent eigenorthogonal decomposition, where n t For the number of time snapshots, The total degrees of freedom; 2) For pulsating fields Apply a positive definite weight matrix with the grid cell area as the diagonal element. Define weighted inner product Perform weighted snapshot POD decomposition; 3) Determine the optimal time eigenfunction ψ by solving the Fredholm integral equation. n (t), retaining the first n based on energy cutoff. m One dominant mode, obtain spatial mode φ n (ξ), time coefficient a n (t) and singular value σ n The solution process satisfies the orthogonality normalization condition: .

3. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 1, characterized in that: S2 includes the following steps: 1) with n m Dimensional POD time coefficient vector Using the input as input, a fully connected residual neural network is constructed after randomly partitioning the dataset, as shown in the following formula: in, For the input embedding layer, N1,…,N L Let P be a residual block and P be the output projection layer; each residual block satisfies , It is an identity operator; 2) Using the mean squared error as the loss function, the formula is as follows: The Adam optimizer is used for training, and batch normalization (BN) and early stopping mechanism are introduced to prevent overfitting. 3) After training converges, the gradient vector is efficiently calculated using inverse mode automatic differentiation (AD), as shown in the following formula: 。 4. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 3, characterized in that: The randomly partitioned dataset in S2 consists of 80% test set, 10% validation set, and 10% test set.

5. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 1, characterized in that, S3 includes the following steps: 1) Construct an estimate of n from gradient samples m ×n m The gradient covariance matrix is ​​given by the following formula: ； 2) Perform eigenvalue decomposition on matrix C, as shown in the following formula: ； 3) Extract the first n feature vectors Zhang Cheng's activity subspace, with For activity variables; 4) Finally, the activity scores for each POD modality are normalized using the following formula: Specifically, sensitivity-driven mode ranking is achieved by comprehensively quantifying the sensitivity contribution of the i-th POD mode in all activity directions.

6. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 1, characterized in that, S4 includes the following steps: 1) Projection operator via active subspace The gradient is projected onto the physical domain to generate a spacetime-sensitive field. in: For the projected gradient vector, and Let it be its k-th component; 2) Statistically analyze the distribution of sensitivity extreme points of all snapshots, and then quantitatively identify the key spatial sites for loss generation and suppression.

7. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 1, characterized in that, S5 includes the following steps: 1) Set n t Projecting each training sample onto an n-dimensional activity subspace yields a set of activity variables. After randomly partitioning the dataset, a polynomial enhanced radial basis function (PRBF) surrogate model is constructed on this low-dimensional space: 2) The kernel function takes the anisotropic squared exponential form: Length scale Determined by maximum likelihood estimation, it is inversely proportional to the j-th eigenvalue of the active subspace; polynomial basis For monomials with a total degree not exceeding d; 3) Apply orthogonal constraints, as shown in the following formula: To ensure well-posedness, the following block linear system is formed: In this case, a small regularization term is added to the diagonal of the kernel matrix K. The weights are then obtained by solving the problem. and ; 4) Passing new input Direct evaluation enables rapid QoI prediction.

8. The parameter sensitivity-driven reduced-order modeling and spatiotemporal attribution method for engineering indicators as described in claim 7, characterized in that: The dataset is randomly divided into 80% training set and 20% test set.

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