Critical heat flux density grey-box prediction method and system considering robustness and physical consistency

Abstract

The present application relates to the technical field of nuclear reactor thermal hydraulic and artificial intelligence, and discloses a critical heat flux density grey box prediction method and system considering robustness and physical consistency, comprising: obtaining a training data set; preprocessing; the input features after preprocessing are respectively subjected to the following operations: substituting into a preset empirical physical model to obtain a physical benchmark prediction value; inputting a neural network to output a correction term; constructing a loss function, and training the neural network using the training data set; after the same preprocessing of the input features of the to-be-predicted working condition, substituting into the empirical physical model and the trained neural network respectively, obtaining the physical benchmark prediction value and the correction term under the to-be-predicted working condition and fusing them to output the final prediction value of the critical heat flux density. The present application takes the empirical physical model as a physical skeleton, and uses the neural network to accurately learn the deviation between the skeleton prediction value and the real experimental value, thereby significantly improving the prediction accuracy of the critical heat flux density.

Description

Technical Field

[0001] This invention relates to the field of nuclear reactor thermal hydraulics and artificial intelligence, specifically to a critical heat flux density ash box prediction method and system that balances robustness and physical consistency. Background Technology

[0002] Critical Heat Flux (CHF) is a core parameter for determining the thermal safety design and operational limits of nuclear reactor cores. Accurate CHF prediction is crucial for ensuring the integrity of nuclear fuel elements and preventing boiling crises. Currently, the nuclear industry widely uses the Groeneveld 2006 Look-Up Table (LUT) as a benchmark tool for CHF prediction. This method is built based on a large amount of experimental data and has good predictive reliability within the normal operating conditions. However, the LUT lookup table method is essentially an interpolation model based on a limited experimental database. When applied to complex operating conditions or extreme geometries (such as small pipe diameters and high pressures), its prediction results often exhibit significant systematic biases, making it difficult to meet high-precision design requirements.

[0003] In recent years, with the development of artificial intelligence technology, purely data-driven machine learning models (such as neural networks and random forests) have been introduced into the field of CHF prediction. These models can directly learn the complex mapping relationship between input parameters (such as pressure, mass flow rate, dryness fraction, and pipe diameter) and CHF from a large amount of experimental data, exhibiting better fitting accuracy than traditional lookup table methods under certain operating conditions. However, purely data-driven models rely entirely on statistical data patterns and lack physical logic constraints; they are essentially "black box" models. In the extrapolation region where training data is sparse, the prediction results are prone to non-physical oscillations, violating basic physical laws (such as CHF monotonically decreasing with increasing dryness fraction), i.e., lacking physical consistency. This contradiction between "accuracy and physical consistency is difficult to balance" constitutes a pressing technical challenge in this field.

[0004] To balance accuracy and physical plausibility, researchers have attempted to combine physical models with data-driven models to form a hybrid modeling approach. For example, Chinese patent CN121980944A discloses a CHF prediction method based on a hybrid mechanism- and data-driven approach. This method employs an additive fusion architecture: "A mechanistic model (such as a LUT) first calculates the CHF baseline value, then trains a neural network to predict the residual (absolute deviation) between the baseline value and the experimental value, and finally adds the two together." This approach belongs to the additive residual correction paradigm, which implicitly assumes that the prediction error of the physical model exhibits a uniform additive shift within the parameter space. This paradigm represents the conventional technical approach used by those skilled in the art when dealing with LUT prediction bias.

[0005] However, through systematic error diagnosis using a large amount of experimental data, the inventors of this application discovered a fundamental deviation between the implicit assumptions of the aforementioned additive residual correction paradigm and the true physical nature of LUT prediction errors. Specifically, through statistical analysis of 3501 sets of CHF experimental data covering pressures of 0.10-20.27 MPa and pipe diameters of 1.02-37.47 mm, a significant positive statistical correlation was found between the absolute prediction error of the LUT2006 lookup table method and the CHF benchmark value (Pearson correlation coefficient r=0.58, p≈0), meaning that the error amplifies proportionally with the CHF magnitude: the average absolute error (659 kW / m²) in the high CHF range (mean approximately 5388 kW / m²) is 5.81 times that in the low CHF range (mean approximately 713 kW / m²) (mean 113 kW / m²). This statistical law clearly reveals the physical nature of LUT errors—not a uniform additive offset, but a multiplicative systematic bias proportional to the benchmark value. This bias can be physically traced back to the boundary effect of the diameter correction formula (D / 8)^-1 / 2 used by the LUT in non-nominal pipe diameter, especially small pipe diameter, operating conditions. When the pipe diameter deviates significantly from the LUT reference pipe diameter (8mm), the deviation introduced by the simplification assumption of this power-law correction formula amplifies as the pipe diameter deviation increases, and it also couples with the magnitude change of the CHF reference value, ultimately exhibiting multiplicative error characteristics.

[0006] It should be noted that although gray box modeling methods have been applied in other engineering fields (such as hydrological prediction, building cooling load prediction, and nonlinear vibration isolation system modeling), these methods address general modeling problems in different engineering scenarios. Their correction targets, bias patterns, and physical mechanisms are completely different from the multiplicative bias of the LUT model in CHF prediction addressed in this application. Cross-domain multiplicative correction schemes target the ratio relationship between different operating conditions or states, rather than correcting the systematic proportional bias of the same physical model under the same input conditions caused by simplification assumptions. The technical problems they address are fundamentally different. Those skilled in the art, when faced with the multiplicative bias problem of the LUT model in CHF prediction, have no motivation to simply transfer gray box modeling schemes from other fields to this field, let alone foresee using a multiplicative correction architecture to match the specific physical nature of multiplicative errors in the CHF field.

[0007] Therefore, there is an urgent need in this field for a prediction method that can match the physical nature of the multiplicative error at the architectural level, ensuring both prediction accuracy and maintaining physical rationality and spatial robustness, thus overcoming the shortcomings of existing technologies. This invention is proposed based on the systematic diagnosis of the multiplicative error. Its core concept lies in not directly predicting the CHF value using a neural network, nor correcting the LUT benchmark value additively. Instead, it uses a neural network to learn a proportional correction factor multiplied by the physical benchmark value, making the correction mechanism naturally match the physical laws of the error at the architectural level. This fundamentally eliminates the multiplicative system bias, achieving a balance between high accuracy, strong physical consistency, and excellent robustness. Summary of the Invention

[0008] This invention provides a critical heat flux density gray box prediction method and system that balances robustness and physical consistency to solve the above-mentioned problems.

[0009] This invention is achieved through the following technical solution:

[0010] A gray-box prediction method for critical heat flux density that balances robustness and physical consistency includes:

[0011] Obtain a training dataset, which contains several sets of input features and corresponding experimental values ​​of critical heat flux density;

[0012] The training dataset is preprocessed;

[0013] The preprocessed input features are subjected to the following operations: (a) substituting them into a preset empirical physical model to obtain a physical baseline prediction value; (b) inputting them into a neural network, which outputs a correction term.

[0014] A loss function is constructed, and the neural network is trained using the preprocessed training dataset. The loss function includes a physical regularization term constructed with the physical benchmark prediction value. The physical regularization term is used to constrain the prediction behavior of the neural network to conform to physical laws.

[0015] After performing the same preprocessing on the input features of the operating condition to be predicted, the values ​​are substituted into the empirical physical model and the trained neural network respectively to obtain the physical baseline prediction value and correction term under the operating condition to be predicted, and then fused to output the final predicted value of the critical heat flux density of the operating condition to be predicted.

[0016] As an optimization, the correction term output by the neural network is a correction coefficient K. Specifically, fusing the physical baseline prediction value and the correction term involves multiplying the physical baseline prediction value by the correction coefficient K. ,in, This is the final predicted value of the critical heat flux density. This is a predicted value based on physical baselines.

[0017] As an optimization, the neural network outputs the natural logarithm lnK of the correction coefficient K, which is calculated by K=exp(lnK).

[0018] As an optimization, the loss function for a neural network whose output is the natural logarithm lnK is:

[0019] ;

[0020] in, For the log-space mean square error term, For globally unbiased terms, For the physical regularization term, and For hyperparameters, Let be the experimental value of the critical heat flux density for the i-th data point. This represents the final predicted critical heat flux density for the i-th training data. Let be the physical baseline prediction value of the i-th training data, n be the number of training data, and std() represent the standard deviation.

[0021] As an optimization, the correction term output by the neural network is the residual value. The fusion of the physical baseline prediction and the correction term specifically involves fusing the physical baseline prediction with the residual value. Add: .

[0022] As an optimization, the output correction term is the residual value. The loss function of the neural network is:

[0023] ;

[0024] in, For globally unbiased terms, For the physical regularization term, For hyperparameters, Let be the experimental value of the critical heat flux density for the i-th data point. This represents the final predicted critical heat flux density for the i-th training data. Let be the physical baseline prediction value of the i-th training data, n be the number of training data, and std() represent the standard deviation.

[0025] As an optimization, the input features include pressure P, mass flow rate G, outlet dryness fraction X, and pipe diameter D; the empirical physical model is the Groeneveld2006 critical heat flux lookup table method; the neural network is a multilayer perceptron containing three hidden layers with 128, 64, and 32 neurons respectively, using the ReLU activation function.

[0026] As an optimization, the preprocessing includes Box-Cox transformation of the experimental values ​​of mass flow rate G and critical heat flux density, and Z-score normalization of all input features.

[0027] This invention also discloses a critical heat flux density gray box prediction system that balances robustness and physical consistency, comprising:

[0028] The data preprocessing module is configured to acquire and preprocess the training dataset, and to perform the same preprocessing on the input features of the working condition to be predicted. The training dataset contains several sets of input features and corresponding experimental values ​​of critical heat flux density.

[0029] The physical baseline calculation module is configured to receive preprocessed input features, substitute them into a preset empirical physical model, and calculate the physical baseline prediction value.

[0030] A neural network correction module includes a neural network that takes preprocessed input features as input and outputs a correction term.

[0031] The loss function calculation module is configured to construct a loss function to train the neural network, wherein the loss function includes a physical regularization term constructed by the physical benchmark prediction value;

[0032] The fusion output module is configured to fuse the physical baseline prediction value corresponding to the same set of input features with the correction term, and output the final predicted value of the critical heat flux density.

[0033] As an optimization, the correction term output by the neural network correction module is a correction coefficient K, and the fusion output module performs multiplicative fusion. Alternatively, the correction term output by the neural network correction module is a residual value. The fusion output module performs additive fusion. .

[0034] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0035] This invention constructs a physics-guided gray-box model, using an empirical physics model as an indispensable physical framework. It then leverages neural networks to accurately learn the deviation between the predicted values ​​and actual experimental values ​​from this framework, significantly improving the prediction accuracy of critical heat flux (CHF). Compared to traditional empirical physics models and purely data-driven models, the predicted values ​​of this invention show a higher degree of agreement with experimental values. Furthermore, the mean, variance, and percentage of samples within the error band of the prediction error are all significantly superior to existing methods, meeting the engineering requirements for CHF prediction accuracy in nuclear reactor thermal safety design.

[0036] Traditional data-driven models rely entirely on statistical patterns and lack physical constraints. They are prone to non-physical oscillations in extrapolation regions with sparse training data, violating fundamental physical laws. This invention uses an empirical physical model as an indispensable "physical framework," ensuring that prediction results are always based on physically reasonable values. Simultaneously, by defining a physical regularization term in the loss function, prior physical knowledge is embedded as an explicit constraint into the neural network training process, guiding the model's prediction behavior in sparse data regions to remain true to physical laws. Experimental verification shows that the prediction results of this invention strictly adhere to key physical laws such as "critical heat flux density monotonically decreases with increasing outlet dryness," a guarantee that pure black-box models cannot guarantee.

[0037] This invention maintains stable and accurate prediction performance even under sparse data or extreme operating conditions, effectively correcting the trend bias caused by the simplification assumptions of traditional empirical physical models. Specifically, in regions with sparse training samples and significant boundary effects of physical model applicability, such as small-diameter pipes and high pressure, the fluctuation range of prediction error is significantly lower than that of traditional methods, demonstrating superior full-parameter space stability compared to traditional methods.

[0038] Traditional pure data-driven models are complete "black boxes," with their internal decision-making processes unpredictable and low reliability in engineering applications. This invention employs a gray-box architecture, explicitly separating the physical model from the data-driven correction terms mathematically. The neural network learns only the correction terms (the correction coefficient K in the multiplicative architecture and the residual in the additive architecture). The correction terms themselves have clear physical meanings. By analyzing the distribution and trends of the correction terms, we can clearly understand in which operating conditions and in what ways the neural network corrects the original physical model, thus transforming it from a "black box" to a "transparent gray box," making it easier for engineers to understand and trust the model's prediction results.

[0039] Existing additive residual correction methods treat all prediction biases of the physical model as additive offsets, failing to fundamentally eliminate multiplicative system bias. This invention, based on a systematic diagnosis of the physical nature of the error, reveals the proportional relationship between the prediction error of the empirical physical model and the benchmark value, and proposes a multiplicative correction coefficient architecture (GB-Mult). This architecture uses a neural network to learn a proportional correction factor K that is multiplied by the physical benchmark value. This correction mechanism naturally aligns with the physical laws of the error at the architectural level, eliminating multiplicative system bias at its source—something existing additive residual correction paradigms cannot achieve.

[0040] In summary, this invention achieves for the first time a unified approach in the field of critical heat flux density prediction, characterized by high prediction accuracy, strong physical consistency, excellent extrapolation robustness, and good engineering interpretability. It overcomes the contradiction between "insufficient accuracy of empirical models" and "poor physical consistency of pure data-driven models" in existing technologies, providing reliable technical support for the thermal safety design and operation limit setting of nuclear reactors. Attached Figure Description

[0041] To more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be considered as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort. In the drawings:

[0042] Figure 1 This is a schematic diagram of the process for a critical heat flux density gray box prediction method that balances robustness and physical consistency, as described in this invention.

[0043] Figure 2 The diagram shows two physical information-guided gray box model architectures proposed in this invention, where (a) is the GB-Add additive residual correction path and (b) is the GB-Mult multiplication correction coefficient path.

[0044] Figure 3 This is a physical monotonicity verification curve of CHF prediction value as a function of export dryness fraction X, used to verify that the prediction results of the model of this invention strictly follow the physical law that "CHF decreases monotonically as export dryness fraction X increases".

[0045] Figure 4 The diagnostic analysis diagram for the multiplicative characteristics of LUT prediction errors includes: (a) the distribution characteristics of absolute error as a function of CHF_LUT; (b) the distribution characteristics of relative error as a function of CHF_LUT; (c) a comparison of the average absolute error across different CHF magnitude ranges; and (d) a comparison of the relative errors of LUTs across different pipe diameter ranges, used to reveal the diameter correction formula. The applicable boundary effect;

[0046] Figure 5 This is a scatter plot comparing the predicted values ​​and experimental values ​​of the GB-Mult model of this invention, used to demonstrate the prediction accuracy of the model on the test set.

[0047] Figure 6 A comparison chart showing the distribution trend of the relative error of different models in the parameter space (pipe diameter D and pressure P) is used to verify the robustness of the model of the present invention in the full parameter space, especially in the small pipe diameter region.

[0048] Figure 7For the residual term of the GB-Add model The distribution and trend plots are used to demonstrate the ability and interpretability of the additive correction architecture in compensating for systematic biases in the physical model;

[0049] Figure 8 This is a distribution and trend graph of the multiplicative correction coefficient K in the GB-Mult model, used to demonstrate the mechanism and interpretability of the multiplicative correction architecture in intelligent nonlinear optimization of traditional physical formulas. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0051] Example 1 discloses a gray box prediction method for critical heat flux density that balances robustness and physical consistency, based on physical information to guide the gray box neural network to predict critical heat flux density.

[0052] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. The following embodiments are only used to illustrate the present invention and should not be regarded as limiting the scope of the present invention. Those skilled in the art can make reasonable adjustments to the parameters (such as the number of network layers, the number of neurons, the learning rate, hyperparameters, etc.) in the embodiments based on the technical concept of the present invention, and such adjustments will not depart from the protection scope of the present invention.

[0053] The development environment in this embodiment can employ any applicable programming language and deep learning framework, such as Python combined with PyTorch, TensorFlow, or other equivalent frameworks. The following description uses Python and PyTorch as examples, but the implementation of this invention is not limited to this.

[0054] The prediction method of this invention includes five steps, as follows:

[0055] S1. Obtain the training dataset, which contains several sets of input features and corresponding experimental values ​​of critical heat flux density.

[0056] S2. Preprocess the training dataset;

[0057] S3. Perform the following operations on the preprocessed input features respectively: (a) Substitute them into the preset empirical physical model to obtain the physical baseline prediction value; (b) Input the value into the neural network, and the neural network outputs the correction term;

[0058] S4. Construct a loss function and train the neural network using the preprocessed training dataset. The loss function includes a physical regularization term constructed with the physical benchmark prediction value. The physical regularization term is used to constrain the prediction behavior of the neural network so that the prediction behavior conforms to physical laws.

[0059] S5. After performing the same preprocessing on the input features of the operating condition to be predicted, substitute them into the empirical physical model and the trained neural network respectively to obtain the physical baseline prediction value and correction term under the operating condition to be predicted, and fuse them to output the final predicted value of the critical heat flux density of the operating condition to be predicted.

[0060] Next, we will describe the implementation process of each step in detail.

[0061] Step S1 is the data preparation stage.

[0062] First, a training dataset is obtained, comprising four input features: pressure (P), mass flow rate (G), outlet dryness fraction (X), and pipe diameter (D), as well as experimental values ​​for critical heat flux (CHF). The dataset is derived from multiple publicly available authoritative literature and technical reports, ensuring the breadth and authority of the data source. The parameter coverage of the dataset can be determined according to actual training needs, preferably covering a wide range of operating conditions, such as pressure from low to supercritical pressure, mass flow rate from low to high flow rate, outlet dryness fraction from subcooled to saturated, and pipe diameter from small to conventional.

[0063] The specific parameter ranges for each data source can be summarized in a table as evidence of the training data sources and parameter coverage.

[0064] For example, pressure 0.10–20.27 MPa, mass flow rate 21–7,961 kg·m⁻²·s⁻¹, outlet dryness -0.49 to 0.91, pipe diameter 1.02–37.47 mm, see Table 1.

[0065] Table 1 shows the sources and parameter ranges of the constructed CHF dataset, compared with the GroeneveldLUT (2006) database.

[0066] Data source Data points Pressure (P)(MPa) Mass flow rate (G) Export dryness (X)(-) Pipe diameter (D) (mm) CHF Dataset (Summary) 3,501 0.10–20.27 21–7,961 -0.49–0.91 1.02–37.47 124–19,275 Thompsonetal. 2,699 0.10–18.96 27–7,961 -0.49–0.91 1.02–37.46 277–19,275 Williamsetal. 282 2.76–15.17 320–4,725 -0.05–0.91 9.51 369–4,073 Weatherheadetal. 193 13.79 171–2,712 -0.36–0.70 7.72 817–5,363 Kimetal. 228 0.11–0.95 21–251 0.32–0.90 12.00 124–1,526 Beusetal. 70 8.27–15.51 400–3,711 -0.02–0.70 7.69 429–3,155 FujioInasakaetal 6 0.31–0.91 4300–11500 -0.115–-0.051 3 7300–16300 Nafeyetal 23 10.1325–20.265 594.44–5361.11 0–0.335 10 674.54–4175.17 LUT(2006) database >24,000 0.1–20.0 0–8,000 -0.5–1.0 1.0–40.0 50–20,000 Coverage assessment - Full coverage Core coverage Full coverage Core coverage Core coverage

[0067] Next, we proceed to step S2, which involves preprocessing the training dataset. This includes the following operations:

[0068] (1) Box-Cox transformation: The experimental values ​​of mass flow rate (G) and critical heat flux density (CHF) are subjected to Box-Cox transformation to correct their possible right-skewed distribution and improve the stability and convergence speed of model training. The transformation parameters can be determined by methods such as maximum likelihood estimation to make the transformed data distribution close to the normal distribution.

[0069] (2) Z-score standardization: All input features (P, G, XD) are subjected to Z-score standardization to transform them into a standard normal distribution with a mean of 0 and a standard deviation of 1. The standardization calculation formula is as follows: ,in and These are the mean and standard deviation of each feature on the training set, respectively. This step aims to eliminate the influence of different physical units on the model's gradient updates.

[0070] Finally, the entire dataset is randomly divided into a training set and a test set according to a certain ratio (such as 8:2 or 7:3). The training set is used for building the neural network and optimizing its parameters, while the test set is used to assess the generalization performance of the neural network.

[0071] Step S3 involves calculating the physical baseline value and diagnosing errors, while simultaneously constructing a gray-box neural network.

[0072] First, for any combination of input features X=[P,G,X,D] in the training and test sets, a pre-defined empirical physics model is invoked to calculate the corresponding physical baseline prediction value. The empirical physical model can employ the Groeneveld 2006 critical heat flux lookup table (LUT) or other known empirical CHF relationships. Taking the Groeneveld 2006 LUT as an example, firstly, the CHF value for a reference pipe diameter (e.g., 8 mm) is obtained by interpolating P, G, and X in the LUT. Then, it is corrected according to the diameter correction formula recommended by the LUT, ultimately yielding the CHF value. This step provides a reliable physical framework for the subsequent gray-box neural network.

[0073] Diagnostic analysis includes the following three levels: (1) Error magnitude statistics, examining the distribution characteristics of absolute and relative errors; (2) Correlation analysis, testing whether the absolute error is correlated with the magnitude of the physical reference predicted value (if significantly correlated, it indicates that the error has multiplicative / proportional characteristics, rather than uniform additive shift); (3) Region-specific analysis, examining the error in different pipe diameter regions (especially the diameter correction formula). Do the non-nominal areas exhibit different behavioral patterns?

[0074] If there is a statistically positive correlation between the absolute error and the physical reference value, that is, the error amplifies proportionally as the magnitude of the reference value increases, it indicates that the prediction error has a multiplicative (proportional) characteristic. The quantitative judgment criteria are: the correlation strength is measured by the Pearson correlation coefficient, and the degree of multiplication is measured by the average error ratio in different magnitude intervals. If the correlation is significant and the error ratio is greater than 2, then the error is determined to be essentially a multiplicative system bias.

[0075] Before constructing the gray-box neural network, a systematic diagnostic analysis was first conducted on the prediction error of the LUT 2006 lookup table method on all 3500 sets of training data to determine the physical nature of the error and provide a physical basis for the selection of the correction paradigm.

[0076] Diagnostic results as follows Figure 4 As shown. Key findings include: the absolute prediction error of the LUT. There is a strong positive correlation with the CHF benchmark value (Pearson r=0.58), and the error increases monotonically with the magnitude of CHF. The average error for the lowest CHF quintile (mean 713 kW / m²) is 113 kW / m², while the average error for the highest CHF quintile (mean 5388 kW / m²) reaches 659 kW / m², the latter being 5.81 times that of the former. This statistical pattern clearly indicates that the prediction error of the LUT has a significant multiplicative (proportional) characteristic, that is, the error tends to be proportional to the benchmark value, rather than exhibiting a uniform additive shift.

[0077] Figure 4 (a) represents the absolute error and A scatter plot of the relationship between them, with LUT predictions The horizontal axis represents the absolute error. - The vertical axis is denoted by . The data points spread out in a fan shape, with errors concentrated in the low CHF region and diverging in the high CHF region. It can be seen that the magnitude of the absolute error increases with the increase of the CHF reference value, indicating that the error has a multiplicative (proportional) characteristic.

[0078] Figure 4 (b) represents the relative error and A scatter plot of the relationship between them, The horizontal axis represents the relative error (%), and the vertical axis represents the relative error (%). The data points are banded across the entire CHF range, uniformly distributed, with no obvious trend. It can be seen that the magnitude of the relative error is independent of the CHF benchmark value, confirming the proportional deviation nature of the error.

[0079] Figure 4 (c) A bar chart representing the average |absolute error| across different CHF magnitude ranges, arranging 3500 data sets according to... The values ​​are divided into 5 intervals (Q1~Q5), and a bar chart is used to display the average |absolute error| for each interval, with the CHF mean value for each interval marked. Key data is as follows: Q1 (mean 713 kW / m²) error 113 kW / m² → Q5 (mean 5388 kW / m²) error 659 kW / m², the latter being 5.81 times that of the former, showing a significant monotonically increasing trend.

[0080] Figure 4(d) Bar chart of relative error of LUT for different tube diameter ranges of gemstones, divided into 5 groups according to tube diameter D (<5, 5–8, 8–10, 10–15, >15 mm). The bar chart (including error bars) shows the mean and standard deviation of the relative error of LUT prediction for each group. It can be seen that the error fluctuation is the largest in the small diameter region (D<8 mm), revealing the LUT diameter correction formula. The applicable boundary effect.

[0081] Next, a multilayer perceptron (MLP) is constructed. The number of neurons in the input layer of this MLP corresponds to the number of input features (e.g., 4, corresponding to the four features P, G, X, and D). The number of hidden layers and the number of neurons per layer can be adjusted according to the actual data scale and model complexity, for example, set to 3 hidden layers with 128, 64, and 32 neurons respectively, or other configurations with different numbers of layers and neurons. Each hidden layer uses the ReLU activation function to introduce non-linearity. The output layer has 1 neuron.

[0082] In this embodiment, the output layer of the neural network is used to predict the natural logarithm lnK of the correction coefficient K. The reason for using lnK instead of directly outputting K is as follows:

[0083] (1) Naturally guaranteeing K>0: CHF is a physical positive quantity, and the correction coefficient K must always be greater than 0. The output lnK of the neural network can take all real numbers. After exponential operation K=exp(lnK), it can be naturally guaranteed that K>0. There is no need to introduce activation functions such as ReLU and Softplus for non-negativity constraints, which reduces model constraints and training complexity.

[0084] (2) Adapting to the multiplicative correction structure and optimizing gradient propagation: The multiplicative correction formula is as follows: Mapped to the logarithmic space as MLPs can directly output lnK, which can participate in linear addition operations in logarithmic space, resulting in continuous gradients and smooth propagation. If the network directly outputs the original K, an additional logarithmic transformation is required before training, increasing the computational layers and easily causing gradient degradation.

[0085] (3) Higher output freedom: the range of values ​​for lnK is One-to-one correspondence with the effective physical interval of K It covers all correction scenarios, resulting in higher training tolerance and fitting freedom.

[0086] The final CHF prediction is obtained by multiplying the physical baseline prediction by the correction factor K: ,in, This is the final predicted value of the critical heat flux density. This is a predicted value based on physical baselines.

[0087] As an alternative implementation, the output layer of the neural network can also be used to predict residual values. At this point, the final CHF prediction value is obtained by adding the physical baseline prediction value to the residual value: .

[0088] Step S4 involves constructing a custom physical loss function and training the model, such as... Figure 2 As shown.

[0089] A custom composite loss function is used to train the constructed gray-box neural network. The training uses a custom loss function that introduces constraint terms on top of basic loss terms such as mean squared error to enforce the elimination of systematic biases in prediction and enhance the physical robustness of the model in sparse data regions.

[0090] For the multiplicative correction architecture (GB-Mult, output lnK), the following composite logarithmic loss function is used, such as Figure 2 As shown in (b):

[0091] ;

[0092] The components are defined as follows:

[0093] (1) Log-space mean square error term This is used to ensure the accuracy of local fitting for each sample:

[0094] ;

[0095] (2) Globally unbiased terms This is used to eliminate global systematic biases in the model.

[0096] ;

[0097] (3) Physical regularization term This value represents the log-standard deviation of the prediction error of the empirical physics model on the training set, signifying the uncertainty of prior physical knowledge. This value is pre-calculated based on the training set before training and remains unchanged during training.

[0098] ;

[0099] in, Let be the final predicted value of the critical heat flux density for the i-th training sample. Let be the experimental value of the critical heat flux density for the i-th training sample. Let be the physical baseline prediction value of the i-th training sample, n be the total number of training samples, and std(·) represent the standard deviation. and This is a hyperparameter used to balance local accuracy, global unbiasedness, and the strength of physical constraints. It can be selected and adjusted through methods such as grid search or Bayesian optimization. The loss function is obtained through... The parameters explicitly balance two objectives: sample-by-sample local accuracy and global statistical unbiasedness. Furthermore, all three objectives need to be jointly optimized through grid search to determine the hyperparameters. and .

[0100] For the additive correction architecture (GB-Add, output residual) Its loss function is, for example Figure 2 As shown in (a):

[0101] ;

[0102] in, For globally unbiased terms, For physical regularization, These are hyperparameters. Grid search optimization is employed to balance data fit with physical prior trust. The loss function aims for global statistical unbiasedness, balancing physical prior trust through λ. The model lacks explicit constraints on per-sample prediction accuracy (only an MSE stabilizing term with a weight of 0.001 is added during training to aid numerical convergence).

[0103] Training configurations can employ conventional optimization algorithms (such as Adam, SGD, etc.), with the initial learning rate set within an appropriate range (e.g., 1×10⁻³ to 1×10⁻). 4 (Scale). Cross-validation strategies (such as 5-fold cross-validation) can be used to adjust hyperparameters and prevent overfitting. Simultaneously, an early stopping mechanism can be introduced, automatically terminating training and restoring the model parameters with the minimum validation set loss when the validation set loss no longer decreases over a certain number of consecutive training epochs.

[0104] While both architectures' loss functions formally embody the physics-guided design philosophy, it's crucial to note that GB-Add and GB-Mult's loss functions differ fundamentally in structural composition, optimization objectives, and training dynamics. The multiplicative correction training scheme cannot be derived from the additive correction scheme through simple reasoning or equivalent substitution. Therefore, the design of the multiplicative correction loss function differs fundamentally from that of the additive correction scheme.

[0105] (1) Different structural composition: The loss function of additive correction is (Two components: globally unbiased + physically regularized), while multiplication is corrected to (Three components: local precision + global unbiasedness + physical regularization).

[0106] (2) The roles of the MSE term are fundamentally different: In additive correction, the original space MSE serves only as an auxiliary stabilizing term (with a minimal weight attached), and is not the main component of the loss function. However, in multiplicative correction, the logarithmic space... It is the core component of the loss function (weights) This is essential because, under the multiplicative correction architecture, the lnK output of the neural network needs to be correlated with the logarithmic space. Adding them together, if no sample-by-sample precision constraint is imposed on the logarithmic space, when When spanning multiple orders of magnitude (e.g., from 124 to 19,275 kW / m²), the fitting residuals in the large baseline region dominate gradient updates in the original space, leading to systematic underfitting in the small baseline region. Therefore, multiplicative correction must establish a local accuracy guarantee mechanism in the logarithmic space, a requirement that does not exist in additive correction.

[0107] (3) The dimensions of the hyperparameter search space are different: Additive correction only needs to optimize a single hyperparameter. (Grid search yields 4 candidate values); while multiplication correction requires joint optimization. and Two hyperparameters are used to find the optimal balance between local accuracy, global unbiasedness, and physical prior trust. The configuration of these hyperparameters directly affects the predicted distribution of the correction coefficient K. If the value is too large, it will ignore the global unbiasedness and lead to systematic bias. If the value is too small, local accuracy will be insufficient, and this trade-off has no counterpart in the additive paradigm. In summary, the loss function design of the multiplicative correction in this invention is not an equivalent replacement of the additive correction, but a systematic redesign based on the order-of-magnitude sensitivity problem and multi-objective optimization requirements unique to the multiplicative architecture.

[0108] Table 2 shows a comparison between the two architectures of this invention and existing technologies:

[0109] Table 2

[0110]

[0111] Step S5 involves making predictions and analyzing the results.

[0112] After performing the same preprocessing on the input features of the work condition to be predicted as on the training data (i.e., the same Box-Cox transformation and Z-score normalization parameters), the following two parallel computation channels are executed respectively:

[0113] (a) Substitute the empirical physical model into the model to obtain the physical baseline prediction value under this working condition. ;

[0114] (b) Input the input into the trained neural network to obtain the correction term (lnK or ...) under this operating condition. ).

[0115] The physical baseline predictions are fused with the correction terms (multiplicative fusion). Where K = exp(lnK); or additive fusion Output the final predicted value of the critical heat flux density under this operating condition. .

[0116] The input features of the test set are preprocessed in the same way and then fed into the trained neural network to obtain the prediction results. The coefficient of determination (COP) is then calculated. The accuracy of the model's predictions is evaluated using metrics such as root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and the percentage of samples whose predicted values ​​fall within the acceptable error range (e.g., ±15%, ±20%).

[0117] like Figure 3 As shown, under the condition of fixing other input features (pressure P, mass flow rate G, pipe diameter D), only the outlet dryness fraction X is changed. The corresponding input features are input into the trained gray box model, and the curve of CHF prediction value changing with outlet dryness fraction X is plotted to verify whether it strictly follows the physical law that "CHF decreases monotonically with the increase of outlet dryness fraction X" in order to verify the physical rationality of the model prediction results.

[0118] Specifically, Figure 3 (a) is a graph showing the relationship between CHF and dryness fraction X (P≈7MPa, G≈1500kg / m²s, D≈10mm). It illustrates the trend of CHF predictions as a function of X for four models (experimental values, LUT, GB-Mult, GB-Add) under fixed pressure, mass flow rate, and pipe diameter, with only the outlet dryness fraction X varying. The graph uses a combination of scatter plots and line graphs to compare the trends. It can be seen that all models show a monotonically decreasing trend of CHF as X increases, while GB-Mult best matches the experimental values.

[0119] Figure 3 (b) is a graph showing the relationship between the CHF mean and dryness index X (statistical analysis by bin for the entire dataset). It shows that all test data were divided into 19 equal intervals according to dryness index X, and the CHF prediction mean of each model within each interval was calculated and compared using a line graph. The bottom is labeled with the physical law: Dryness index X↑→CHF".

[0120] The robustness of the model can be verified in the following way: input features from the test set in sparse or extreme operating conditions such as small-diameter areas and high-pressure areas are input into the trained gray box model and the traditional empirical physical model (such as LUT), respectively, and the prediction error distribution of the two in this area is compared; if the error of the gray box model is significantly smaller and the fluctuation is more stable, it proves that it has better robustness.

[0121] The prediction results show that the model constructed in this embodiment achieves excellent prediction performance on the test set, and all evaluation indicators are superior to traditional empirical physical models and pure data-driven models. Furthermore, the model predictions strictly follow key physical laws and exhibit good robustness under extreme conditions, fully verifying the effectiveness of the method proposed in this invention.

[0122] After undergoing the same preprocessing, the input features of the test set are fed into the trained GB-Mult model (neural network) to obtain prediction results. The results show that the model constructed in this embodiment achieves excellent performance on the test set: a high coefficient of determination (R²) of 0.9664, a low mean absolute percentage error (MAPE) of 9.89%, and 76.75% of the sample prediction errors falling within the engineering-acceptable ±15% error band.

[0123] like Figure 5 As shown, the model's predicted values ​​are in high agreement with the experimental values, and the data points are closely distributed on both sides of the diagonal of y=x, proving its extremely high prediction accuracy.

[0124] Figure 5 Contains 5 subgraphs (e.g.) Figure 5 As shown (two at the top, three at the bottom), these plots display scatter plots of predicted CHF versus experimental CHF for LUT 2006, MLP, RandomForest, GB-Mult, and GB-Add, respectively. Each subplot is labeled with the y=x diagonal, ±15% error boundary line, R², and RMSE.

[0125] All subplots use a uniform coordinate axis scale and square proportion, facilitating horizontal comparison of the accuracy of different models. Figure 1 (Top left) LUT 2006: the baseline empirical physical model; sub- Figure 2 (Top right) is MLP: Pure black-box neural network (MSE loss); sub- Figure 3 (Bottom left) RandomForest: a pure data-driven ensemble model; sub- Figure 4 (Bottom middle) is GB-Mult: Multiplication Corrected Gray Box Model (this invention); sub- Figure 5 (Bottom right) is the GB-Add: Additive Correction Gray Box Model.

[0126] like Figure 6 As shown, the model exhibits stability in the full parameter space, especially in the small diameter (D<10mm) region, which is not present in the traditional LUT method. It effectively corrects the trend bias caused by the simplification of the diameter correction formula in the latter, demonstrating strong robustness.

[0127] Figure 6(a) is a scatter plot showing the relationship between relative error and pipe diameter D. Specifically, it shows the scatter plot distribution of the relative error of the LUT, MLP, and GB-Mult models as a function of pipe diameter D. The ±15% error band is marked with green semi-transparent, and the small diameter region (D<5mm) is marked with yellow. It can be seen that LUT has a significant positive bias (overestimation) in the small diameter region; MLP has a large error dispersion; and GB-Mult's error is concentrated in the ±15% band across the entire pipe diameter range.

[0128] Figure 6 (b) is a scatter plot of the relative error and pressure P, as above, with the horizontal axis representing pressure P (MPa). It can be seen that GB-Mult has the most concentrated error across the entire pressure range, exhibiting the best parameter space robustness.

[0129] Corrections for gray box architecture (such as residuals) or coefficient The model itself has physical meaning and can analyze the improvements made by the original physical model, realizing the transformation from a "black box" to a "transparent gray box," such as... Figure 7 , Figure 8 As shown.

[0130] Specifically, Figure 7 (a) is the residual correction term for GB-Add. Distribution histogram, residual values ​​output by the GB-Add model The probability density distribution histogram is overlaid with the KDE curve. (Annotation) Reference line, mean line, and statistical information box (Mean, Std, upward / downward adjustment ratio). The distribution approximates a zero-mean normal, indicating that the additive correction symmetrically adjusts the LUT predictions with zero as the center.

[0131] Figure 7 (b) is GB-Add And pipe diameter D (system deviation capture), A scatter plot of pipe diameter D (colors mapped to actual CHF values), overlaid. The trend line of the mean changing with D (red broken line). (Annotation) The line and small diameter area (D<5mm) are marked in yellow. It can be seen that... In the small-diameter region, a systematic negative bias (downward adjustment of LUT) is observed. The red trend line visually demonstrates the compensation law of additive correction for the systematic bias of LUT.

[0132] Figure 8(a) is the histogram of the distribution of the correction coefficient K in GB-Mult, the probability density distribution histogram of the correction coefficient K output by the GB-Mult model, and the KDE curve superimposed. The reference line for K=1 (no correction), the mean line, and the statistical information box (K>1 adjustment ratio, K<1 adjustment ratio) are labeled. It can be seen that the K values ​​are concentrated around 1, exhibiting an approximately symmetrical distribution, and the multiplicative correction does not introduce any abnormal amplification or reduction.

[0133] Figure 8 (b) is a trend chart of K and pipe diameter D (physical interpretability) in GB-Mult, a scatter plot of K versus pipe diameter D (color mapping to actual CHF values), and an overlay of a trend line showing the change of the mean K with D. The K=1 line and the small-diameter region are marked. The red trend line shows the evolution of the mean K from small-diameter K<1 (down-adjusting LUT) to large-diameter K≈1. It can be seen that K is systematically <1 (down-adjusted) in the small-diameter region. The monotonic change of the trend line reflects the physical interpretability of the correction coefficient K. The neural network intelligently corrects the over-extension of the LUT diameter correction formula under small-diameter conditions through K.

[0134] In summary, this embodiment verifies the significant effect of the proposed method in improving the accuracy, physical consistency, and robustness of critical heat flux density prediction.

[0135] Example 2 discloses a critical heat flux density gray box prediction system that balances robustness and physical consistency, comprising:

[0136] The data preprocessing module is configured to acquire and preprocess the training dataset, and to perform the same preprocessing on the input features of the working condition to be predicted. The training dataset contains several sets of input features and corresponding experimental values ​​of critical heat flux density.

[0137] The physical baseline calculation module is configured to receive preprocessed input features, substitute them into a preset empirical physical model, and calculate the physical baseline prediction value.

[0138] A neural network correction module includes a neural network that takes preprocessed input features as input and outputs a correction term.

[0139] The loss function calculation module is configured to construct a loss function to train the neural network, wherein the loss function includes a physical regularization term constructed by the physical benchmark prediction value;

[0140] The fusion output module is configured to fuse the physical baseline prediction value corresponding to the same set of input features with the correction term, and output the final predicted value of the critical heat flux density.

[0141] In some embodiments, the correction term output by the neural network correction module is a correction coefficient K, and the fusion output module performs multiplicative fusion. Alternatively, the correction term output by the neural network correction module is a residual value. The fusion output module performs additive fusion. .

[0142] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A gray box prediction method for critical heat flux density that balances robustness and physical consistency, characterized in that, include: Obtain a training dataset, which contains several sets of input features and corresponding experimental values ​​of critical heat flux density; The training dataset is preprocessed; The preprocessed input features are subjected to the following operations: (a) substituting them into a preset empirical physical model to obtain a physical baseline prediction value; (b) inputting them into a neural network, which outputs a correction term. A loss function is constructed, and the neural network is trained using the preprocessed training dataset. The loss function includes a physical regularization term constructed with the physical benchmark prediction value. The physical regularization term is used to constrain the prediction behavior of the neural network to conform to physical laws. After performing the same preprocessing on the input features of the operating condition to be predicted, the values ​​are substituted into the empirical physical model and the trained neural network respectively to obtain the physical baseline prediction value and correction term under the operating condition to be predicted, and then fused to output the final predicted value of the critical heat flux density of the operating condition to be predicted.

2. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 1, characterized in that, The correction term output by the neural network is a correction coefficient K. Specifically, fusing the physical baseline prediction value and the correction term involves multiplying the physical baseline prediction value by the correction coefficient K. ,in, This is the final predicted value of the critical heat flux density. This is a predicted value based on physical baselines.

3. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 2, characterized in that, The neural network outputs the natural logarithm lnK of the correction coefficient K, which is calculated by K=exp(lnK).

4. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 2, characterized in that, The loss function for a neural network whose output is the natural logarithm lnK is: ； in, For the log-space mean square error term, For globally unbiased terms, For the physical regularization term, and For hyperparameters, Let be the experimental value of the critical heat flux density for the i-th data point. This represents the final predicted critical heat flux density for the i-th training data. Let be the physical baseline prediction value for the i-th training data, n be the number of training data, and std() represent the standard deviation.

5. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 1, characterized in that, The correction term output by the neural network is the residual value. The fusion of the physical baseline prediction and the correction term specifically involves fusing the physical baseline prediction with the residual value. Add: .

6. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 5, characterized in that, The output correction term is the residual value. The loss function of the neural network is: ； in, For globally unbiased terms, For the physical regularization term, For hyperparameters, Let be the experimental value of the critical heat flux density for the i-th data point. This represents the final predicted critical heat flux density for the i-th training data. Let be the physical baseline prediction value for the i-th training data, n be the number of training data, and std() represent the standard deviation.

7. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 1, characterized in that, The input features include pressure P, mass flow rate G, outlet dryness fraction X, and pipe diameter D; the empirical physical model is the Groeneveld2006 critical heat flux lookup table method; the neural network is a multilayer perceptron containing three hidden layers with 128, 64, and 32 neurons respectively, using the ReLU activation function.

8. The critical heat flux density gray box prediction method that balances robustness and physical consistency according to claim 1, characterized in that, The preprocessing includes performing Box-Cox transformation on the experimental values ​​of mass flow rate G and critical heat flux density, and Z-score normalization on all input features.

9. A critical heat flux density gray box prediction system that balances robustness and physical consistency, characterized in that, include: The data preprocessing module is configured to acquire and preprocess the training dataset, and to perform the same preprocessing on the input features of the working condition to be predicted. The training dataset contains several sets of input features and corresponding experimental values ​​of critical heat flux density. The physical baseline calculation module is configured to receive preprocessed input features, substitute them into a preset empirical physical model, and calculate the physical baseline prediction value. A neural network correction module includes a neural network that takes preprocessed input features as input and outputs a correction term. The loss function calculation module is configured to construct a loss function to train the neural network, wherein the loss function includes a physical regularization term constructed by the physical benchmark prediction value; The fusion output module is configured to fuse the physical baseline prediction value corresponding to the same set of input features with the correction term, and output the final predicted value of the critical heat flux density.

10. A critical heat flux density gray box prediction system that balances robustness and physical consistency according to claim 9, characterized in that, The correction term output by the neural network correction module is the correction coefficient K, and the fusion output module performs multiplicative fusion. ; Alternatively, the correction term output by the neural network correction module is a residual value. The fusion output module performs additive fusion. .

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