Supersonic cascade flow field reconstruction method fusing transfer learning theory and adaptive dynamic physical partitioning

Abstract

The invention discloses a supersonic cascade flow field reconstruction method fusing a transfer learning theory and adaptive dynamic physical partitioning, and belongs to the technical field of fluid mechanics numerical simulation and machine learning. According to the method, through a physical partitioning strategy, the problem of feature learning fragmentation caused by lack of physical logic in pure data partitioning is avoided; by means of modularized cooperative training of a shared feature extractor and a partition adapter, the defect that a global model is insufficient in local difference learning is overcome, and redundancy of repeated learning of universal features when partition models are independently trained is avoided; in the prediction and iterative optimization link, through automatic verification and backtracking adjustment, the defects of high manual intervention cost, non-physical phenomenon of a prediction result and large local partition error are avoided, and finally efficient, accurate, physical and reasonable prediction of high-precision data is realized.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of fluid mechanics numerical simulation and machine learning, which realizes the prediction and iterative optimization of complex flow field under new working conditions by fusing high-precision and low-precision data and combining physical laws with deep learning models. BACKGROUND

[0002] In modern aerospace, mechanical engineering and fluid mechanics, cascade is a core component of key components such as aircraft engines and gas turbines. Accurate simulation and prediction of its flow field characteristics are crucial. However, traditional flow field simulation methods have many limitations: traditional numerical simulation methods often face high computational cost, sensitivity to boundary conditions, and difficulty in capturing subtle physical phenomena when dealing with complex flow fields. In terms of multi-dimensional data fusion, there are significant differences in accuracy, spatial resolution and physical dimension between two-dimensional and three-dimensional simulation data, making it difficult to directly compare and utilize the data. For flow field prediction under new working conditions, existing methods often rely on empirical formulas or simplified assumptions, lacking deep learning and generalization ability for flow field physical laws, resulting in insufficient prediction accuracy. In addition, traditional methods lack targeted model optimization strategies when dealing with key physical regions in the flow field, making it difficult to adapt to complex flow field changes under different working conditions.

[0003] In summary, in order to overcome the shortcomings of traditional methods, there is an urgent need for a new method that can fuse multi-dimensional data, deeply mine flow field physical laws, and have efficient prediction and optimization capabilities. SUMMARY

[0004] To solve the problems existing in the prior art, the present application provides a supersonic cascade flow field reconstruction method that fuses transfer learning theory and self-adaptive dynamic physical partition, to deeply mine flow field physical laws and fuse multi-dimensional data, improve prediction efficiency and optimization capability.

[0005] The technical solution adopted to achieve the purpose of the present application is: a supersonic cascade flow field reconstruction method that fuses transfer learning theory and self-adaptive dynamic physical partition, comprising the following steps:

[0006] Step 1: Preprocess the obtained high-precision and low-precision data sets.

[0007] Step 2: Regularize each physical partition of the high-precision and low-precision data sets.

[0008] Step 3: Jointly train to build a shared feature extractor and a partition adapter collaborative optimization model.

[0009] Step 4: Based on the trained model, input unknown working condition parameters for prediction, and calculate errors for iterative optimization of the model.

[0010] The method specifically comprises the following steps:

[0011] Step 1: Based on the cascade entity geometric parameters, a full-size three-dimensional parameterized model is constructed; after completing the topological decomposition and structured grid division of the fluid domain, boundary conditions, initial field and turbulence model parameters are set in the CFD solver, and numerical calculation is performed to obtain a high-precision data set. The low-precision data set is obtained according to the cascade size to construct a two-dimensional model, and the same boundary conditions, initial field and turbulence model parameters as the three-dimensional calculation are set for two-dimensional calculation.

[0012] Step 2: The high and low precision data sets are preprocessed. Since the simulation data has irregular shape characteristics, the feature processor cannot learn it, so the data needs to be filled to a regular shape to facilitate information extraction by the feature extractor. The filling value area is an invalid information area, and the invalid area needs to be masked. The mask formula is:

[0013]

[0014] The effective area physical quantity is normalized to map the density gradient and other physical quantities to the [0, 1] interval. The normalization formula is:

[0015]

[0016] Wherein, The normalized physical quantity is represented by The original physical quantity is represented by And The minimum and maximum values of the physical quantity are represented by

[0017] Step 3: Physical partitioning based on K-means unsupervised adaptive method. Since the density gradient size under different working conditions of high and low precision data is not uniform, the density gradient under each working condition needs to be adaptively physically partitioned and adjusted to a uniform fixed size. The formula for the K-means unsupervised adaptive method is:

[0018]

[0019]

[0020]

[0021] Wherein, The k-th cluster centroid is represented by The cardinality in the cluster is represented by The inverse of the cluster size is represented by The sum of all data point vectors in the cluster is represented by , The point set that meets the conditions in the curly braces is represented by a set of all data points satisfy the following conditions, squared distance from less than or equal to its distance from .

[0022] Step 4: Joint training, build a shared feature extractor, use a convolutional neural network (CNN) to extract common features from low-precision and high-precision data in all partitions. The convolution operation of the CNN can be represented as:

[0023]

[0024] where, represents the output feature map after convolution operation, represents the convolution kernel, represents the input feature map, represents the bias term, represents the activation function, represents the convolution operation.

[0025] Design an independent partition adapter for each physical partition, which uses a neural network to further process the common features extracted by the shared feature extractor to learn the differences between two-dimensional features and three-dimensional features. The neural network of the partition adapter can be represented as:

[0026]

[0027] where, represents the output of the partition adapter, represents the input feature, represents the parameters of the neural network, represents the mapping function of the neural network.

[0028] In the joint training phase, in order to enable the model to simultaneously learn the general flow field rules across partitions and the low-precision to high-precision data differences specific to each partition, the parameters of the shared feature extractor and all partition adapters need to be optimized collaboratively. The specific process is as follows:

[0029] Step 4.1: Build a loss function, which is the core basis for optimizing model parameters. Build a loss function that considers the shared feature extractor and partition adapter. This loss function consists of two parts: the loss term of the shared feature extractor and the loss term of all partition adapters. The mathematical expression is as follows:

[0030]

[0031] where: Loss of shared feature extractor, which measures the difference between the universal features extracted by the shared feature extractor and the real universal flow field rules. Loss of the th partition adapter, which measures the difference between the prediction results of the th partition adapter and the real three-dimensional features. Number of partition adapters. and are weight coefficients, which are used to balance the proportion of the loss of the shared feature extractor and the partition adapter in the total loss, and their optimal values are usually determined by cross-validation and other methods.

[0032] Step 4.2: Loss calculation of shared feature extractor, the loss of the shared feature extractor is calculated by comparing the universal features extracted by the shared feature extractor with the corresponding universal features extracted from high-precision data. Assuming that the extracted universal features are , and the corresponding universal features extracted from high-precision data are , then can be represented as:

[0033]

[0034] where: represents the number of training samples; represents the universal features extracted by the shared feature extractor corresponding to the th training sample; represents the universal features extracted from high-precision data corresponding to the th training sample. represents the square of the Euclidean norm, which is used to measure the difference between the features.

[0035] Step 4.3: Loss calculation of partition adapter, the loss of each partition adapter is calculated by comparing the prediction results of the partition adapter with the real three-dimensional features. Assuming that the prediction results of the th partition adapter for the th training sample are , and the corresponding real three-dimensional features are , then can be represented as:

[0036]

[0037] where: represents the prediction results of the th partition adapter for the th training sample. represents the the real three-dimensional features of the i-th partition corresponding to the i-th training sample.

[0038] Step 4.4: The parameters are updated by the back propagation method. Before starting the optimization, the parameters of the shared feature extractor and all partition adapters are initialized. Usually, a random initialization method is used to ensure that the initial parameters have a certain randomness, avoiding the model falling into a local optimum. The training data is input into the model, and the general features are extracted by the shared feature extractor. Then the general features are input into each partition adapter to obtain the prediction results of each partition. According to the loss function constructed above, the total loss under the current model parameters is calculated . Starting from the loss function, the gradients of each parameter in the model to the loss are calculated in turn according to the chain rule. For the parameters of the shared feature extractor, the gradient calculation expression is:

[0039]

[0040] Update parameters: According to the calculated gradient, the model parameters are updated using the optimization method. Taking the stochastic gradient descent method as an example, the parameter update formula is:

[0041]

[0042] where is the learning rate, which controls the step size of parameter update.

[0043] Step 4.5: Iterative training until convergence. Before training begins, the conditions for model convergence are preset. The loss function value changes less than a certain threshold in consecutive multiple training periods, or the error on the validation set no longer decreases significantly. In a complete training period, all samples in the training set are used to train the model, and the training loss is calculated. At the same time, the model performance is evaluated on the validation set, and the validation loss is calculated. According to the validation loss, it is judged whether the model begins to overfit. If it is found that the loss decreases slowly or oscillates during training, the learning rate decay strategy can be used to gradually reduce the learning rate, improving the stability of model convergence. During the training process, whenever the loss on the validation set reaches a new minimum value, the current model parameters are saved as the optimal model. After training is completed, the final optimal model obtained is the model after collaborative optimization, at which time the prediction error of the model has stabilized within an acceptable range.

[0044] ​Step 5: prediction and iterative optimization, input new working condition parameters, extract and predict new working condition features combined with shared feature extractor for each partition adapter, generate high-precision results for the partition. The prediction results of each partition are spliced into a complete output result, and the splicing place is smoothed. Iterative optimization verification is performed, including boundary smoothing verification and local partition error verification. If the splicing trace or error is too large, adjust the splicing strategy or refine the partition boundary, and retrain the model until the error meets the preset standard.

[0045] The beneficial effects of the present application are: the method can fine modeling according to the characteristics of different physical partitions through the innovative partition independent prediction and iterative optimization strategy, avoiding the "one-size-fits-all" mode of traditional methods when dealing with complex flow field, thereby greatly improving the accuracy and adaptability of prediction. In addition, the method realizes the collaborative optimization of shared feature extractor and partition adapter through joint training mechanism, not only effectively reduces the consumption of computing resources, but also enhances the generalization ability of the model to new working conditions, reduces the dependence on a large amount of experimental data. This combination of partition prediction and global optimization not only ensures the prediction accuracy, but also greatly improves the running efficiency, providing an efficient, flexible and reliable solution for flow field prediction in the fields of aerospace, mechanical engineering and other fields. The method significantly improves the efficiency and accuracy of the method in cascade flow field prediction.

[0046] The method of the present application avoids the fragmentation of feature learning caused by the lack of physical logic through the physical partition strategy; with the help of modular collaborative training of shared feature extractor and partition adapter, it not only avoids the defects of global model learning insufficient local differences, but also avoids the redundancy of repeated learning of general features when training each partition model separately; in the prediction and iterative optimization link, through automatic verification and backtracking adjustment, it avoids the high cost of manual intervention and the drawbacks of non-physical phenomena and large local partition error in the prediction result, and finally realizes efficient, accurate and physically reasonable prediction of high-precision data. BRIEF DESCRIPTION OF DRAWINGS

[0047] Figure 1 is a flowchart of the method.

[0048] Figure 2 is a density gradient cloud chart of the filled cascade.

[0049] Figure 3 is a mask operation diagram for the filled cascade region.

[0050] Figure 4 is a partition diagram of cascade density gradient at 1.6Ma.

[0051] Figure 5 is a schematic diagram of the original density gradient of the cascade at 1.395Ma.

[0052] Figure 6 is a schematic diagram of predicted 1.395Ma cascade density gradient.

[0053] Figure 7 is a schematic diagram of relative error between predicted and actual. DETAILED DESCRIPTION

[0054] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0055] Embodiment 1

[0056] The present application proposes a supersonic cascade flow field reconstruction method fusing the theories of transfer learning and adaptive dynamic physical partition, comprising the following steps:

[0057] Step 1: Based on the cascade entity geometric parameters, a full-size three-dimensional parameterized model is constructed; after completing the topological decomposition and structured grid division of the fluid domain, the boundary conditions, initial field and turbulence model parameters are set in the CFD solver, and numerical calculation is performed to obtain a high-precision data set. The low-precision data set is obtained according to the cascade size to construct a two-dimensional model, and the same boundary conditions, initial field and turbulence model parameters as the three-dimensional calculation are set for two-dimensional calculation.

[0058] Step 2: High-precision data of 0.7~1.6Ma interval 0.05 Mach number working condition and low-precision data of 0.7~1.7Ma interval 0.01 Mach number working condition are preprocessed, since the simulation data shape has irregular characteristics and the feature processor cannot learn it, the data needs to be filled to a regular shape to facilitate the feature extractor to extract information. The filling value area is an invalid information area, and the invalid area needs to be masked, and the mask formula is:

[0059]

[0060] The effective area physical quantity is normalized, and the density gradient and other physical quantities are mapped to the [0,1] interval, and the normalization formula is:

[0061]

[0062] wherein, denotes the normalized physical quantity, denotes the original physical quantity, and respectively denote the minimum value and the maximum value of the physical quantity.

[0063] Step 3: Physical partitioning based on K-means unsupervised adaptive method. Since the density gradient size under different working conditions of high and low precision data is not uniform, it is necessary to adaptively partition the density gradient under each working condition to adjust it to a uniform fixed size. The formula for the K-means unsupervised adaptive method is:

[0064]

[0065]

[0066]

[0067] where, denotes the centroid of the k-th cluster, denotes the cardinality of the cluster, denotes the inverse of the cluster size , the sum of all data point vectors in the cluster , denotes the set of points that satisfy the conditions in the curly braces, denotes the set of all data points satisfies the following conditions, squared distance from is less than or equal to its distance from .

[0068] Step 4: Joint training, build a shared feature extractor, use a convolutional neural network (CNN) to extract common features from all partitioned low-precision and high-precision data. The convolution operation of the CNN can be represented as:

[0069]

[0070] where, denotes the output feature map after convolution operation, denotes the convolution kernel, denotes the input feature map, denotes the bias term, denotes the activation function, denotes the convolution operation.

[0071] Design an independent partition adapter for each physical partition, which uses a neural network to further process the common features extracted by the shared feature extractor to learn the differences between two-dimensional features and three-dimensional features. The neural network of the partition adapter can be represented as:

[0072]

[0073] wherein, represents the output of the partition adapter, represents the input feature, represents the parameter of the neural network, represents the mapping function of the neural network.

[0074] In the joint training phase, in order to enable the model to simultaneously learn the general flow field law across partitions and the low-to-high precision data difference specific to each partition, the parameters of the shared feature extractor and all partition adapters need to be collaboratively optimized. The specific process is as follows:

[0075] Step 4.1: Construct the loss function, which is the core basis for optimizing model parameters. A loss function that comprehensively considers the shared feature extractor and the partition adapter is constructed. The loss function consists of two parts: the loss term of the shared feature extractor, and the loss term of all partition adapters. The mathematical expression is as follows:

[0076]

[0077] wherein, represents the loss of the shared feature extractor, which measures the difference between the general features extracted by the shared feature extractor and the real general flow field law. represents the loss of the th partition adapter, which measures the difference between the prediction result of the th partition adapter and the real three-dimensional feature. represents the number of partition adapters. and are weight coefficients, used to balance the proportion of the loss of the shared feature extractor and the partition adapter in the total loss, and their optimal values are usually determined by cross-validation and other methods.

[0078] Step 4.2: Loss calculation of the shared feature extractor, the loss of the shared feature extractor is calculated by comparing the general features extracted by the shared feature extractor with the corresponding general features extracted from high-precision data. Assuming that the extracted general features are , and the corresponding general features extracted from high-precision data are , then can be represented as:

[0079]

[0080] wherein, represents the number of training samples; represents the general feature extracted by the shared feature extractor corresponding to the th training sample; represents the general feature extracted from high-precision data corresponding to the The general feature extracted from the high-precision data corresponding to the training sample. represents the square of the Euclidean norm, used to measure the difference between features.

[0081] Step 4.3: Loss calculation of partition adapter, loss of each partition adapter is calculated by comparing the prediction result of the partition adapter with the real three-dimensional feature. Assuming that the prediction result of the th partition adapter for the th training sample is , and the corresponding real three-dimensional feature is , then can be expressed as:

[0082]

[0083] wherein: represents the prediction result of the th partition adapter for the th training sample. represents the real three-dimensional feature of the th partition corresponding to the th training sample.

[0084] Step 4.4: Update parameters by backpropagation method, initialize the parameters of the shared feature extractor and all partition adapters before starting optimization. Usually, random initialization method is adopted to ensure that the initial parameters have certain randomness and avoid the model falling into local optimum. Input the training data into the model, extract the general feature through the shared feature extractor, then input the general feature into each partition adapter to get the prediction result of each partition. According to the loss function constructed above, calculate the total loss under the current model parameters. According to the chain rule, calculate the gradient of each parameter in the model to the loss . For the parameters of the shared feature extractor, the gradient calculation expression is:

[0085]

[0086] Update parameters: according to the calculated gradient, update the model parameters using optimization method. Taking the stochastic gradient descent method as an example, the parameter update formula is:

[0087]

[0088] wherein, represents the learning rate, which controls the step size of parameter update.

[0089] Step 4.5: Iterative training until convergence, before training starts, preset the conditions for model convergence, the loss function value changes less than a certain threshold in continuous multiple training cycles, or the error on the validation set no longer decreases significantly. In a complete training cycle, all samples in the training set are used to train the model, and the training loss is calculated. At the same time, the model performance is evaluated on the validation set, and the validation loss is calculated. According to the validation loss, it is judged whether the model starts to overfit. If it is found that the loss decreases slowly or fluctuates during training, learning rate decay strategy can be used to gradually reduce the learning rate to improve the stability of model convergence. During the training process, whenever the loss on the validation set reaches a new minimum value, the current model parameters are saved as the optimal model. After training is completed, the final optimal model obtained is the model after collaborative optimization, and at this time the prediction error of the model has stabilized in an acceptable range.

[0090] Step 5: Prediction and iterative optimization, input new working condition 1.395Ma, extract and predict the new working condition features of each partition adapter combined with the shared feature extractor to generate high-precision results for the partition. The prediction results of each partition are spliced into complete output results, and the splicing part is smoothed. Iterative optimization verification is performed, including boundary smoothing verification and local partition error verification. Finally, the prediction result under the working condition of 1.395Ma is obtained, and the relative error with the high-precision data is 3.92%.

[0091] The above is a further detailed description of the present application in combination with a specific preferred technical solution, and cannot be regarded as limiting the specific implementation of the present application to these descriptions. For ordinary skilled persons in the technical field to which the present application belongs, without departing from the concept of the present application, a number of simple deductions or substitutions can be made, which should be regarded as falling within the protection scope of the present application.

[0092] The above shows and describes the basic principles and main features of the present application and the advantages of the present application, and it is obvious for those skilled in the art that the present application is not limited to the details of the above exemplary embodiments, and the present application can be realized in other specific forms without departing from the spirit or basic characteristics of the present application. Therefore, from any point of view, the embodiments should be regarded as exemplary and non-limiting, and the scope of the present application is defined by the appended claims rather than the above description, and therefore all changes falling within the meaning and scope of the equivalent elements of the claims are intended to be included in the present application. Any reference signs in the claims should not be regarded as limiting the claims involved.

Claims

1. A method for reconstructing the flow field of a supersonic blade cascade by integrating transfer learning theory and adaptive dynamic physical partitioning, characterized in that, Includes the following steps: Step 1: Preprocess the high and low precision datasets by filling irregularly shaped simulation data into regular shapes, filling numerical regions as invalid information regions, and performing masking. The physical quantities in the effective region are normalized, and physical quantities such as density gradient are mapped to the [0,1] interval; Step 2: The density gradient magnitudes of high- and low-precision data are not uniform under different working conditions. We physically partition the data based on the K-means unsupervised adaptive method and adjust it to a uniform fixed size. Step 3: Joint training to build a shared feature extractor, using a convolutional neural network (CNN) to extract common features from high- and low-precision data across all physical partitions; Each physical partition is configured with an independent partition adapter, which uses a neural network to further process the common features extracted by the shared feature extractor to learn the difference between two-dimensional features and three-dimensional features. During the joint training phase, the parameters of the shared feature extractor and all partition adapters are collaboratively optimized, enabling the model to simultaneously learn the general flow field patterns across partitions and the low-to-high-precision data differences specific to each partition; the specific process is as follows: Step 3.1: Construct the loss function, which consists of two parts: the loss term for the shared feature extractor and the loss term for all partition adapters; Step 3.2: Calculation of the loss of the shared feature extractor. It is calculated by comparing the general features extracted by the shared feature extractor with the corresponding general features extracted from the high-precision data; Step 3.3: Loss calculation for partition adapters, loss for each partition adapter. It is calculated by comparing the prediction results of the partition adapter with the actual 3D features; Step 3.4: Update parameters using the backpropagation method. Before starting optimization, initialize the parameters of the shared feature extractor and all partition adapters; input the training data into the model, extract common features through the shared feature extractor, and then input the common features into each partition adapter to obtain the prediction result for each partition; calculate the total loss under the current model parameters based on the loss function constructed above. Starting from the loss function, calculate the impact of each parameter in the model on the loss in turn according to the chain rule. The gradient is calculated; then, based on the calculated gradient, the model parameters are updated using optimization methods. Step 3.5: Iterate training until convergence; Step 4: Prediction and Iterative Optimization. Input new operating condition parameters, extract and predict new operating condition features for each partition adapter in conjunction with the shared feature extractor, and generate high-precision results for that partition; stitch the prediction results of each partition into a complete output result, and smooth the stitching area; perform iterative optimization verification, including boundary smoothing verification and local partition error verification; if stitching traces or excessive errors are found, adjust the stitching strategy or refine the partition boundaries, and retrain the model until the error meets the preset standard.

2. The method according to claim 1, characterized in that: In step 1, the formula for masking is: ； The normalization formula is: ； in, This represents the normalized physical quantity. Represents the original physical quantity. and These represent the minimum and maximum values ​​of a physical quantity, respectively.

3. The method according to claim 1, characterized in that: In step 2, the formula for the K-means unsupervised adaptive method is: ； ； ； in, This represents the centroid of the k-th cluster. Indicates the cardinality in the cluster. The reciprocal of the cluster size The sum of all data point vectors in the cluster , This represents the set of points that satisfy the conditions within the curly braces. Represents the set of all data points The following conditions must be met. Square distance since Less than or equal to its .

4. The method according to claim 1, characterized in that: In step 3, the convolution operation of CNN is represented as follows: ； in, This represents the output feature map after the convolution operation. Represents the convolution kernel. Indicates the input feature map, Indicates the bias term. This represents the activation function. Indicates the convolution operation; The neural network representation of the partition adapter is as follows: ； in, This indicates the output of the partition adapter. Indicates input features, The parameters of the neural network, This represents the mapping function of the neural network.

5. The method according to claim 1, characterized in that: In step 3.1, the mathematical expression of the loss function is as follows: ； in: This represents the loss of the shared feature extractor, used to measure the difference between the general features extracted by the shared feature extractor and the true general flow field laws; Indicates the first The loss of the first partition adapter is used to measure the loss of the second partition adapter. The difference between the prediction results of each partition adapter and the true 3D features; Indicates the number of partition adapters; and These are weighting coefficients used to balance the proportion of the loss from the shared feature extractor and the partitioning adapter in the total loss.

6. The method according to claim 1, characterized in that: In step 3.2, the loss calculation of the shared feature extractor is specifically as follows: the extracted general features are... The corresponding general features extracted from high-precision data are ,but Represented as: ； in: Indicates the number of training samples; Indicates the first The common features extracted by the shared feature extractor corresponding to each training sample; Indicates the first The general features extracted from high-precision data correspond to each training sample; It represents the square of the Euclidean norm, used to measure the difference between features.

7. The method according to claim 1, characterized in that: In step 3.3, the loss calculation of the partition adapter is specifically as follows: assuming the first... The partition adapter for the first... The prediction results for each training sample are The corresponding true three-dimensional features are ,but Represented as: ； in: express The partition adapter for the first... Prediction results for each training sample; Indicates the first The training sample corresponding to the th training sample The true three-dimensional features of each partition.

8. The method according to claim 1, characterized in that, In step 3.5, the parameters of the shared feature extractor are... Its gradient calculation expression is: ； The formula for updating the parameters is: ； in, This represents the learning rate, which controls the step size for updating parameters.

9. The method according to claim 1, characterized in that: Step 3.5 specifically involves: before training begins, setting the convergence conditions for the model, where the loss function value changes less than a certain threshold over multiple consecutive training cycles or the error on the validation set no longer decreases significantly; within a complete training cycle, training the model using all samples in the training set and calculating the training loss; simultaneously, evaluating the model performance on the validation set and calculating the validation loss. The validation loss is used to determine whether the model is starting to overfit. If the loss decreases slowly or oscillates during training, a learning rate decay strategy is adopted to gradually reduce the learning rate. During training, whenever the loss on the validation set reaches a new minimum value, the current model parameters are saved as the optimal model. After training is completed, the final optimal model is the co-optimized model.

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