Neural network mechanism information embedding method based on dimension-by-dimension sample enhancement and latent variable space sampling
By combining dimensional sample augmentation and latent variable space sampling with a mechanism-constrained loss function, the problem of embedding mechanism information in neural network models is solved, achieving efficient generalization and improved interpretability under unknown conditions.
Patent Information
- Application Number
- CN202511265043.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-05
- Publication Date
- 2025-12-23
Smart Images
Figure CN121189415A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of mechanism and data fusion driven modeling, specifically involving a method for embedding neural network mechanism information based on dimensional sample augmentation and latent variable space sampling. Background Technology
[0002] Accurate modeling is a crucial technological foundation for modern industrial process state prediction, fault diagnosis, and optimized control; model accuracy directly impacts the effectiveness of intelligent applications. Existing modeling methods can be broadly categorized into mechanistic modeling and data-driven modeling. Mechanistic modeling, based on first-principles calculations, characterizes processes such as flow, heat transfer, and chemical reactions through mathematical equations. However, for complex industrial engineering, systems contain numerous unknown and unmeasurable parameters, posing a significant challenge to accurate modeling. Furthermore, mechanistic models typically rely on solving complex differential equations, resulting in low computational efficiency and difficulty in meeting the requirements of real-time prediction or monitoring. In contrast, data-driven modeling does not require in-depth understanding of the system's complex internal mechanisms; it can directly extract the correlation characteristics between system parameters from historical data. These methods offer high fitting accuracy and fast computation speed, making them suitable for online monitoring and optimization. However, the accuracy of data-driven models depends on the density and coverage of training samples. When operating conditions exceed the distribution of training data, problems such as weak generalization ability and decreased accuracy can arise, making it difficult to guarantee prediction accuracy and stability. Secondly, data-driven models are essentially "black box" models, and their poor interpretability is due to drawbacks such as the difficulty in explaining their internal mechanisms and the lack of clear physical meaning.
[0003] Given the advantages and disadvantages of both mechanism-driven and data-driven methods, existing research attempts to combine mechanistic knowledge with machine learning, proposing a mechanism-data fusion-driven method to achieve complementary advantages between the two methods: on the one hand, prior mechanism knowledge can improve the generalization and interpretability of the model; on the other hand, data-driven methods can achieve flexible and efficient modeling and real-time model response. Currently, methods such as mechanistic information neural networks convert mechanism knowledge into equality or inequality constraints and embed them into the neural network loss function to achieve effective fusion of mechanism and data, which has received widespread attention in fields such as flow, heat transfer, materials, and medicine. However, these methods still have the following problems: (1) the form of embedding mechanism information into the neural network is not yet clear; (2) the scope of the mechanism constraints is affected by the distribution of training samples, and the interpretability of the model under unknown working conditions cannot be effectively guaranteed. Summary of the Invention
[0004] Purpose of the invention: The purpose of this invention is to provide a method for embedding neural network mechanism information based on dimensional sample augmentation and latent variable space sampling, which realizes the efficient embedding of mechanism information in neural network models and can effectively guarantee the generalization ability and interpretability of the model even in working conditions outside the coverage of training samples.
[0005] Technical solution: The present invention provides a method for embedding neural network mechanism information based on dimensional sample augmentation and latent variable space sampling, comprising:
[0006] Normalize the given original small sample dataset; after setting the augmentation range of each variable distribution interval, augment the original samples in the preprocessed original small sample dataset dimension by dimension, and obtain a large-scale augmented sample set after sampling within the augmentation interval; the preprocessed original small sample dataset and the large-scale augmented sample set form a training sample set.
[0007] The ITNN neural network model is called and trained based on a training sample set. During the training process, the input samples and network weights are iteratively updated to obtain a trained model.
[0008] The latent variable distribution space is constructed based on the boundary conditions of the input samples of the primary training model, and uniform sampling is performed within the latent variable distribution space to obtain the latent variable synthetic sample set; the latent variable synthetic sample set and the preprocessed original small sample dataset are combined to form the secondary training sample set.
[0009] A mathematical representation of the relationship between each input variable is constructed, forming a mechanism-constrained loss and embedding it into the loss function of the primary training model. The primary training model is then trained using a secondary training sample set, ultimately yielding a mechanism-input neural network model that embeds prior mechanism information.
[0010] Furthermore, the given original small sample dataset undergoes normalization preprocessing, including:
[0011] Given a small original sample dataset, perform max-min normalization to map it to the standard interval [-1, 1] to obtain the original normalized sample set X∈R. N×M ;
[0012]
[0013] Furthermore, after setting the enhancement magnitude for each variable's distribution interval, the original samples in the preprocessed original small sample dataset are enhanced dimension-by-dimensionally. After sampling within the enhancement interval, a large-scale enhanced sample set is obtained, including:
[0014] Set variables i = 1, 2, ..., M, and initialize variable i = 1;
[0015] Based on the original normalized sample set X∈R N×M Given the standard interval [-1, 1], the sample augmentation interval for variable i is set as follows: in, and These represent the magnitude of the enhancement of the original upper and lower limits of variable i, respectively;
[0016] Let variable i be the number of samples to be augmented, and let N be the number of samples to be augmented. (i) ,from N is obtained by continuous uniform sampling within the interval (i) By creating a single-dimensional sample and setting the data for the other variables to be empty, we obtain an augmented sample set for variable i.
[0017] If i = i + 1, and i > M, proceed to the next step; otherwise, reset the sample augmentation interval of variable i.
[0018] Ultimately, the augmented sample sets of the M groups of variables form the augmented sample set X. aug ={X aug,(i)}, where i = 1, 2, ..., M.
[0019] Furthermore, the ITNN neural network model is invoked and trained based on a training sample set. During the training process, the input samples and network weights are iteratively updated simultaneously to obtain a trained model, including:
[0020] The original normalized sample set X and the enhanced sample set X aug As a training sample for the ITNN neural network model;
[0021] The loss function of the ITNN neural network model is set as follows:
[0022]
[0023] in, The reconstruction loss is the original normalized sample set. Enhance the one-dimensional reconstruction loss of sample i; λ i for The corresponding weighting coefficients;
[0024] A trained model is obtained by simultaneously updating the input variables and network weights through gradient descent.
[0025] Furthermore, the reconstruction loss of the original normalized sample set And the one-dimensional reconstruction loss of the enhanced sample with variable i Its expression is as follows:
[0026]
[0027] Where, x ji Let i be the original value of variable i in the j-th original normalized sample; The model output value of variable i in the j-th original normalized sample; Let i be the augmentation value of variable i in the j-th augmented sample; Let i be the sum of variable i in the j-th augmented sample and the model output value.
[0028] Furthermore, the ITNN neural network model includes an input layer, several hidden layers, and an output layer, wherein the number of nodes in the input layer is less than the number of nodes in the output layer.
[0029] Furthermore, a latent variable distribution space is constructed based on the boundary conditions of the input samples of the first-time trained model, and uniform sampling is performed within the latent variable distribution space to obtain a synthetic latent variable sample set, including:
[0030] Let the input variables be i = 1, 2, ..., l, the initial variable be i = 1, and the number of samples to be sampled be N. syn ;
[0031] Find the minimum value of input variable i in the latent variable distribution space. and maximum value In the interval N is obtained by continuous uniform sampling within. syn Group of single-dimensional synthetic samples u syn,(i) ;
[0032] If i = i + 1, and i > 1, then proceed to the next step; otherwise, reset the latent variable.
[0033] Space arrangement;
[0034] Finally, the latent variable synthetic sample set U is obtained. syn ={u syn,(i)}, i = 1, 2, ..., l.
[0035] Furthermore, a mathematical representation of the relationship between each input variable is constructed, forming a mechanistic constraint loss and embedding it into the loss function of the primary training model. The primary training model is then trained using a secondary training sample set, ultimately yielding a mechanistic information input neural network model embedding prior mechanistic information, including:
[0036] The secondary training model uses the input variables and network weights as initialization parameters;
[0037] Original normalized sample set X and latent variable synthetic sample set U syn As secondary training samples, the secondary training samples are used to train the primary training model;
[0038] The loss function is set as follows:
[0039]
[0040] in, The reconstruction loss is the original normalized sample set. The k-th mechanistic loss function is used for the second training sample; K is the number of mechanistic loss functions.
[0041] Gradient descent is used to update both the input variables and the network weights simultaneously.
[0042] Furthermore, the reconstruction loss of the original normalized sample set Its expression is as follows:
[0043]
[0044] Where, x ji Let i be the original value of variable i in the j-th original normalized sample; Let i be the model output value of variable i in the j-th original normalized sample.
[0045] Furthermore, the k-th mechanistic loss of the secondary training samples Its expression is as follows:
[0046]
[0047] Where, N syn f is the number of samples to be sampled; (k) (·) represents the function of the k-th algebraic equation; Let i be the model output value of variable i in the nth sample, where i = 1, 2, ..., n.
[0048] Beneficial effects: Compared with the prior art, the significant technical effects of the present invention are as follows: It adopts a dimension-wise sample augmentation method based on the original small sample dataset and a latent variable space uniform sampling strategy based on the boundary conditions of the training samples, which solves the problem that the model output violates the mechanism information and the model interpretability is poor under unknown working conditions due to insufficient sample quantity, and finally realizes the effective embedding of mechanism information into the neural network framework. Attached Figure Description
[0049] Figure 1 This is a schematic diagram of the process of the present invention;
[0050] Figure 2 A schematic diagram of the data structure for a single training sample set;
[0051] Figure 3 A schematic diagram of the input training neural network structure;
[0052] Figure 4 This is a schematic diagram of the data distribution of the original sample set;
[0053] Figure 5 Scatter plots showing the training and testing results of the corresponding models under different methods. Detailed Implementation
[0054] The technical solution of the present invention will now be described in conjunction with specific embodiments and the accompanying drawings.
[0055] like Figure 1 As shown, the present invention provides a method for embedding neural network mechanism information based on dimensional sample augmentation and latent variable space sampling, comprising the following steps:
[0056] S1. Normalize the given original small sample dataset; after setting the enhancement range of each variable distribution interval, enhance the original samples in the preprocessed original small sample dataset dimension by dimension, and obtain a large-scale enhanced sample set after sampling within the enhancement interval; the preprocessed original small sample dataset and the large-scale enhanced sample set form a training sample set.
[0057] The specific implementation process of step S1 is as follows:
[0058] S1.1 Perform normalization preprocessing on the given original small sample dataset, including:
[0059] Given a small original sample dataset, perform max-min normalization to map it to the standard interval [-1, 1] to obtain the original normalized sample set X∈R. N×M ;
[0060]
[0061] S1.2 After setting the enhancement range for each variable's distribution interval, the original samples in the preprocessed original small sample dataset are enhanced dimension by dimension. After sampling within the enhancement interval, a large-scale enhanced sample set is obtained, including:
[0062] In this embodiment, the original normalized sample set X∈R N×M Perform dimension-by-dimensional sample augmentation. The specific augmentation steps are as follows:
[0063] S1.2.1. Set variables i = 1, 2, ..., M, and initialize variable i = 1;
[0064] S1.2.2, Based on the original normalized sample set X∈R N×M Given the standard interval [-1, 1], the sample augmentation interval for variable i is set as follows: in, and These represent the magnitude of the enhancement of the original upper and lower limits of variable i, respectively;
[0065] S1.2.3, Set the number of samples to be augmented for variable i to be N. (i) ,from N is obtained by continuous uniform sampling within the interval (i) By creating a single-dimensional sample and setting the data for the other variables to be empty, we obtain an augmented sample set for variable i.
[0066] S1.2.4, i = i + 1, if i > M, then proceed to step S1.2.5, otherwise proceed to step S1.2.2;
[0067] S1.2.5 Finally, the augmented sample set of the M groups of variables forms the augmented sample set X. aug ={X aug,(i)}, where i = 1, 2, ..., M.
[0068] In this scheme, the enhanced sample set is obtained by performing dimension-wise enhancement and sampling on the standard interval where the original normalized sample is located.
[0069] S1.3, Preprocessed original small sample dataset (i.e., original normalized sample set X) and augmented sample set X aug To form a training sample set, such as Figure 2 As shown.
[0070] S2. Call the ITNN neural network model and train the ITNN neural network model based on a training sample set. During the training process, the input samples and network weights are iteratively updated to obtain a trained model.
[0071] This invention employs an input-trained neural network structure to achieve effective embedding of mechanistic information. For example... Figure 3 As shown, the ITNN neural network model consists of an input layer, several hidden layers, and an output layer. The number of nodes in the input layer is generally less than the number of nodes in the output layer. During training, the model input is treated as an unknown variable and optimized along with the network parameters.
[0072] The specific implementation process of step S2 is as follows:
[0073] S2.1, Combine the original normalized sample set X and the enhanced sample set X aug As a training sample for the ITNN neural network model;
[0074] S2.2, Set the loss function of the ITNN neural network model as follows:
[0075]
[0076] in, The reconstruction loss is the original normalized sample set. Enhance the one-dimensional reconstruction loss of sample i; λi for The corresponding weighting coefficients; and This can be further expressed as:
[0077]
[0078] Where, x ji Let i be the original value of variable i in the j-th original normalized sample; The model output value of variable i in the j-th original normalized sample; Let i be the augmentation value of variable i in the j-th augmented sample; Let i be the sum of variable i in the j-th augmented sample and the model output value.
[0079] S2.3. The input variables and network weights are updated simultaneously through gradient descent to obtain a trained model.
[0080] S3. Construct a latent variable distribution space based on the boundary conditions of the input samples of the primary training model, and perform uniform sampling within the latent variable distribution space to obtain a latent variable synthetic sample set; the latent variable synthetic sample set and the preprocessed original small sample dataset are combined to form a secondary training sample set.
[0081] The latent variable synthesis sample set is obtained by sampling the input variables updated after one training iteration. Specifically, latent variable sampling is performed on the input variables obtained in step S2.3.
[0082] The specific implementation process of step S3 is as follows:
[0083] S3.1. Set the input variables i = 1, 2, ..., l, the initial variable i = 1, and the number of samples to be sampled is N. syn ;
[0084] S3.2 Obtain the minimum value of input variable i in the latent variable distribution space. and maximum value In the interval N is obtained by continuous uniform sampling within. syn Group of single-dimensional synthetic samples u syn,(i) ;
[0085] S3.3, i = i + 1, if i > 1, then proceed to step S3.4, otherwise proceed to step S3.2;
[0086] S3.4 Finally, the latent variable synthetic sample set U is obtained. syn ={u syn,(i)}, i = 1, 2, ..., l.
[0087] S4. Construct a mathematical representation of the relationship between each input variable, form a mechanism constraint loss and embed it into the loss function of the primary training model. Combine the secondary training sample set to train the primary training model, and finally obtain a mechanism information input neural network model with embedded prior mechanism information.
[0088] During the secondary training process, a standardized mathematical representation that can be embedded into a neural network architecture is constructed to achieve the embedding of mechanistic information. The mathematical representation is typically based on the three fundamental laws of conservation of energy, mass, and momentum. The key parameters of the equipment (pressure, temperature, enthalpy, mass flow rate, etc.) strictly follow the mechanisms of action of the fundamental laws of thermodynamics. Common equality constraints include algebraic equations and partial differential equations, while inequality constraints are used to describe the range of variable values, including monotonicity and boundary conditions.
[0089] The secondary training process in step S4 is as follows:
[0090] S4.1 The secondary training model uses the input variables and network weights obtained in step S2 as initialization parameters;
[0091] S4.2, Original normalized sample set X and latent variable composite sample set U syn As secondary training samples for PI-ITNN, the secondary training samples are used to train the primary training model.
[0092] S4.3. Set the loss function as follows:
[0093]
[0094] in, The reconstruction loss is the original normalized sample set. The k-th mechanistic loss function is used for the second training sample; K is the number of mechanistic loss functions. and This can be further expressed as:
[0095]
[0096] In the above formula: x ji Let i be the original value of variable i in the j-th original normalized sample; N represents the model output value of variable i in the j-th original normalized sample; syn f is the number of samples to be sampled; (k) (·) represents the function of the k-th algebraic equation; Let i be the model output value of variable i in the nth sample, where i = 1, 2, ..., n.
[0097] S4.4 Update input variables and network weights simultaneously using gradient descent.
[0098] This invention significantly improves the interpretability and extrapolation generalization performance of neural network models through the deep integration of data-driven approaches and mechanism embedding. The specific implementation process includes the following key steps: First, for the original small sample dataset, a large-scale augmented sample set conforming to the distribution characteristics of the original data is constructed using a dimension-wise augmentation method. Second, the initial neural network model is trained once using the input training neural network structure based on the original samples and augmented samples. Then, adaptive uniform sampling is implemented in the latent variable space of the first-stage training model to effectively fill the sparse regions of the training samples in the feature space. Finally, the original samples and the latent variable composite samples are used together as secondary training samples, where the original samples simultaneously optimize regression loss and mechanism loss, while the latent variable composite samples only participate in the calculation of mechanism loss. This ensures that the model output strictly follows objective laws while maintaining data fitting accuracy. This method achieves reliable modeling of machine learning models in data-scarce scenarios through the synergistic optimization of data augmentation and mechanism embedding.
[0099] The technical solution of this invention uses an Input Training Neural Network (ITNN) as the model basis, trains the model once using a small number of original samples and augmented samples, and then uses a uniform sampling method to generate synthetic samples in the latent variable space of the model trained once. Then, the mechanism constraint loss is embedded in the second training process using the latent variable synthetic samples as carriers, thereby improving the model's generalization ability and interpretability for unknown working conditions.
[0100] The method of this invention will be verified through a mathematical simulation case below.
[0101] Construct a simulation dataset containing 6 variables, the mathematical expression of which is shown below:
[0102]
[0103] In the formula, γ is generated independently from the uniform distribution U(0,3), and e i ~N(0,0.02),i=1,2,3…,6 is random noise that follows a Gaussian distribution.
[0104] The above formula generates 20 sets of imbalanced data as the original sample set, whose distribution is as follows: Figure 4 As shown. Furthermore, 2000 sets of uniformly distributed U(-3,6) data were generated from the above formula as test samples to verify the effectiveness of the method of this patent.
[0105] (1) Mechanism Constraint Construction
[0106] A third-dimensional mechanistic constraint based on algebraic polynomials is constructed for the above six-dimensional dataset. The mechanistic constraint loss is shown in the following equation:
[0107]
[0108] (2) Model and evaluation index setting
[0109] The model is constructed using an ITNN architecture. This model has one node in the input layer, six nodes in the output layer, and two hidden layers, employing the Sigmoid activation function. This patent uses the coefficient of determination (R-squared, R...). 2 Quantitatively evaluate the model's performance:
[0110]
[0111] In the formula: N is the number of samples, and n is the number of sample dimensions. and For the i-th sample, the actual value and the reconstructed value of the j-th variable are... This represents the average value of the j-th variable in the actual data.
[0112] (3) Comparison and analysis of model results before and after embedding mechanism information
[0113] To verify the effectiveness of the proposed mechanistic information embedding method, this patented method was compared with several other methods, as shown in Table 1. To avoid changes in results due to model differences, all methods used models with the same network structure and parameter configuration.
[0114] Table 1 Comparison of embedding mechanisms and dimensional sample augmentation of each method
[0115]
[0116] Following the methods outlined in Table 1, the corresponding models were trained. Considering the impact of synthetic sample selection, each model underwent 50 random sampling iterations to evaluate the robustness of its performance. Figure 5 Scatter plots of model training and testing results under different methods are provided. The closer the scatter points are to the straight line y=x, the better the model training or testing results. Figure 4 It can be seen that the training results of the models corresponding to the three methods are stable. However, when testing the models, the models corresponding to methods 1 and 2 generally perform better within the range covered by the training samples, while the models perform worse outside the range. The model corresponding to method 3 can still maintain good test results outside the range covered by the training samples, which verifies that the method proposed in this patent has certain advantages.
Claims
1. A method for embedding neural network mechanism information based on dimensional sample augmentation and latent variable space sampling, characterized in that, include: Normalize the given original small sample dataset; After setting the enhancement magnitude for each variable's distribution interval, the original samples in the preprocessed original small sample dataset are enhanced dimension by dimension. After sampling within the enhancement interval, the enhanced sample set is obtained. The preprocessed original small sample dataset and the enhanced sample set together form a training sample set. The ITNN neural network model is called and trained based on a training sample set. During the training process, the input samples and network weights are iteratively updated to obtain a trained model. The latent variable distribution space is constructed based on the boundary conditions of the input samples of the first-time training model, and uniform sampling is performed in the latent variable distribution space to obtain the latent variable synthetic sample set; the latent variable synthetic sample set and the preprocessed original small sample dataset are combined to form the secondary training sample set. A mathematical representation of the relationship between each input variable is constructed, forming a mechanism-constrained loss and embedding it into the loss function of the primary training model. The primary training model is then trained using a secondary training sample set, ultimately yielding a mechanism-input neural network model that embeds prior mechanism information.
2. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that, Normalization preprocessing is performed on the given original small sample dataset, including: Given a small original sample dataset, perform max-min normalization to map it to the standard interval [-1, 1] to obtain the original normalized sample set X∈R. N×M ; 3. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that, After setting the enhancement magnitude for each variable's distribution interval, the original samples in the preprocessed original small sample dataset are enhanced dimension-by-dimensionally. Sampling within the enhancement interval yields a large-scale enhanced sample set, including: Set variables i = 1, 2, ..., M, and initialize variable i = 1; Based on the original normalized sample set X∈R N×M Given the standard interval [-1, 1], the sample augmentation interval for variable i is set as follows: in, and These represent the magnitude of the enhancement of the original upper and lower limits of variable i, respectively; Let variable i be the number of samples to be augmented, and let N be the number of samples to be augmented. (i) ,from N is obtained by continuous uniform sampling within the interval (i) By creating a single-dimensional sample and setting the data for the other variables to be empty, we obtain an augmented sample set for variable i. If i = i + 1, and i > M, proceed to the next step; otherwise, reset the sample augmentation interval of variable i. Ultimately, the augmented sample sets of the M groups of variables form the augmented sample set X. aug ={X aug,(i) }, where i = 1, 2, ..., M.
4. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that, The ITNN neural network model is invoked and trained based on a single training sample set. During training, the input samples and network weights are iteratively updated simultaneously to obtain a trained model, including: The original normalized sample set X and the enhanced sample set X aug As a training sample for the ITNN neural network model; The loss function of the ITNN neural network model is set as follows: in, The reconstruction loss is the original normalized sample set. Enhance the one-dimensional reconstruction loss of sample i; λ i for The corresponding weighting coefficients; A trained model is obtained by simultaneously updating the input variables and network weights through gradient descent.
5. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 4, characterized in that, The reconstruction loss of the original normalized sample set And the one-dimensional reconstruction loss of the enhanced sample with variable i Its expression is as follows: Where, x ji Let i be the original value of variable i in the j-th original normalized sample; The model output value of variable i in the j-th original normalized sample; Let i be the augmentation value of variable i in the j-th augmented sample; Let i be the sum of variable i in the j-th augmented sample and the model output value.
6. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that: The ITNN neural network model includes an input layer, several hidden layers, and an output layer, wherein the number of nodes in the input layer is less than the number of nodes in the output layer.
7. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that, A latent variable distribution space is constructed based on the boundary conditions of the input samples of the trained model, and uniform sampling is performed within the latent variable distribution space to obtain a synthetic latent variable sample set, including: Let the input variables be i = 1, 2, ..., l, the initial variable be i = 1, and the number of samples to be sampled be N. syn ; Find the minimum value of input variable i in the latent variable distribution space. and maximum value In the interval N is obtained by continuous uniform sampling within. syn Group of single-dimensional synthetic samples u syn,(i) ; If i = i + 1, and i > 1, then proceed to the next step; otherwise, reset the latent variable distribution space. Finally, the latent variable synthetic sample set U is obtained. syn ={u syn,(i) }, i = 1, 2, ..., l.
8. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 1, characterized in that, A mathematical representation of the relationships between the input variables is constructed, forming a mechanistic constraint loss and embedding it into the loss function of the primary training model. The primary training model is then trained using a secondary training sample set, ultimately yielding a mechanistic information input neural network model embedding prior mechanistic information, including: The secondary training model uses the input variables and network weights as initialization parameters; Original normalized sample set X and latent variable synthetic sample set U syn As secondary training samples, the secondary training samples are used to train the primary training model; The loss function is set as follows: in, The reconstruction loss is the original normalized sample set. The k-th mechanistic loss function is used for the second training sample; K is the number of mechanistic loss functions. Gradient descent is used to update both the input variables and the network weights simultaneously.
9. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 8, characterized in that, The reconstruction loss of the original normalized sample set Its expression is as follows: Where, x ji Let i be the original value of variable i in the j-th original normalized sample; Let i be the model output value of variable i in the j-th original normalized sample.
10. The neural network mechanism information embedding method based on dimensional sample augmentation and latent variable space sampling according to claim 8, characterized in that, The k-th mechanistic loss of the secondary training samples Its expression is as follows: Where, N syn f is the number of samples to be sampled; (k) (·) represents the function of the k-th algebraic equation; Let i be the model output value of variable i in the nth sample, where i = 1, 2, ..., n.