Accidental explosion concrete penetration depth prediction method fused with experience algorithm knowledge

Abstract

The invention combines the advantages of priori knowledge fusion, provides an accidental explosion concrete penetration depth prediction method fused with experience algorithm knowledge, and aims to improve the prediction precision and physical rationality in a small sample scene. The method comprises the following steps: firstly, constructing a combined parameter Z through a physical parameter, a target attribute and a Forrest formula, and taking the combined parameter Z as an input feature of a model to represent a physical rule; then, a multi-objective loss function is designed, the multi-objective loss function comprises a data fitting item, a physical constraint item and a boundary constraint item, the physical constraint item verifies the physical consistency of predicted values through trace disturbance input parameters, and the boundary constraint item ensures that the penetration depth is non-negative. And further constructing a dynamic weight adjustment mechanism, and automatically balancing the optimal weight of the physical law and data fitting according to the training process. And based on a back propagation neural network framework, performing model training in combination with the constraints. According to the method, the mean square error is remarkably reduced in a concrete penetration task, and meanwhile, the physical interpretability of a data-driven model is expanded.

Description

Technical Field

[0001] The present invention relates to the technical field of information processing, and particularly relates to a method for predicting the penetration depth of accidental explosion concrete by integrating empirical algorithm knowledge. Background Art

[0002] In the prediction of explosion damage effects, especially in the process of predicting penetration depth, machine learning methods usually rely on a large amount of training data to capture the complex patterns and laws of explosion effects. However, in the case of limited datasets, traditional machine learning methods are difficult to accurately capture the multi-dimensional impacts during the explosion process, resulting in insufficient prediction accuracy and robustness. Especially in practical applications, the experimental data of many explosion accidents are scarce or difficult to obtain, which makes it difficult to solely rely on data-driven model training. How to incorporate prior knowledge has become the key to the invention.

[0003] In the representation and integration of prior knowledge, how to effectively introduce prior knowledge such as the experience of domain experts, physical laws, and real conditions into the model directly affects the prediction accuracy and reliability of the model. First of all, the challenge in the representation of prior knowledge is how to transform these abstract and theoretical knowledge into a form that can be understood and processed by machine learning models. For example, physical laws may involve complex mathematical relationships, dependencies between variables, and even implicit laws, all of which need to be represented through appropriate representation methods. For numerical prior knowledge, it can be represented by explicit numerical parameters or constraints; for rules or boundary conditions, they can be expressed through logical rules, conditional constraints, etc. In addition, prior knowledge may also include empirical relationships, preferences, or limiting conditions, which are not easy to directly quantify, but can be represented by designing specific structures.

[0004] To solve this problem, a strategy of combining prior knowledge with machine learning has been proposed. The prior knowledge includes physical principles such as the Forrestal formula and existing theoretical formulas. These knowledge provide necessary structural guidance for the model, helping the model better understand the basic laws of explosion effects. Especially when the data is insufficient, prior knowledge can effectively make up for the lack of training data. By introducing prior knowledge, the model can not only improve the prediction accuracy on small datasets, but also enhance its robustness and better conform to the actual physical behavior.

[0005] Specifically, the core problem lies in how to effectively integrate the Forrestal formula with machine learning models. If the existing methods directly add the results of the Forrestal formula to the model, they often face the problem of insufficient generalization ability. Therefore, it is necessary to design appropriate loss functions and constraint conditions so that during the model training process, the model can not only accurately fit the data but also ensure that the model follows physical laws. How to capture the physical characteristics in the empirical formula and integrate them into the model, while balancing data-driven learning and the introduction of prior knowledge, and avoiding over-reliance on certain aspects of knowledge during the model optimization process, is a major challenge faced by this invention. Thus, designing appropriate loss functions and training strategies to achieve an effective balance between data fitting and physical consistency for the model becomes the key point of this invention. Based on the above analysis, this invention proposes an accidental explosion concrete penetration depth prediction method that integrates empirical algorithm knowledge. Summary of the Invention

[0006] Object of the Invention: In practical applications, the experimental data of many explosion accidents are scarce or difficult to obtain, which makes it difficult to solely rely on data-driven model training. Traditional machine learning methods are difficult to accurately capture the multi-dimensional impacts during the explosion process, resulting in insufficient prediction accuracy and robustness. To overcome the above problems, this invention proposes a framework that combines machine learning and prior knowledge, aiming to improve the prediction accuracy and robustness of explosion damage effects. This framework combines physical laws and data-driven learning methods to ensure that the model output conforms to physical laws in the case of insufficient data.

[0007] To achieve the above object, the technical solution adopted by this invention is as follows: An accidental explosion concrete penetration depth prediction method that integrates empirical algorithm knowledge, comprising the following steps:

[0008] (1) Construct the input by integrating physical prior knowledge. Based on the Forrestal formula, analyze the physical relationship between the penetration depth and the fragment velocity, density, shape factor, and the compressive strength of the target, and derive the combined parameter Z as an additional input feature of the model. Then, fuse it with the experimental data and input it into the neural network.

[0009] (1.1) Physical feature analysis and combined parameter generation based on the Forrestal formula.

[0010] (1.1.1) First, analyze the physical formula. According to the analytical solution of the Forrestal formula:

[0011]

[0012] Among them, P represents the penetration depth, and the other input parameters include: d represents the diameter of the fragment, M represents the mass of the fragment, V represents the velocity of the fragment, f represents the compressive strength of the concrete target, pt represents the density of the concrete target, N represents the fragment shape factor (dimensionless), and S represents a dimensionless empirical constant. By integrating these physical quantities to determine the core influencing factors, the model can accurately predict the penetration depth of the fragment during the explosion under different conditions, thereby providing theoretical support for the study of explosion damage effects.

[0013] (1.1.2) Construction of combined parameters. According to the analytical solution of Forrestal, there is a certain relationship between the penetration depth (P) and physical parameters. Specifically, the penetration depth is related to factors such as the target density (p), the fragment shape factor (N), the square of the fragment velocity (V1 2 ), as well as the dimensionless empirical constant (S) and the target compressive strength (f). Define the dimensionless combined parameter and use it as the key physical characteristic representing the interaction strength between the projectile and the target. By analyzing the formula, it can be found that the penetration depth has a positive correlation with Z. This parameter comprehensively quantifies the dynamic balance relationship between kinetic energy and the anti-penetration ability of the target material, providing physical prior guidance for the model. Specifically, the change in the penetration depth P obtained from the Forrestal formula is affected by Z. As Z increases, the penetration depth shows an increasing trend. In addition, although this relationship is not linear but shows asymptotic changes through logarithmic terms, in theoretical analysis, the positive correlation between the penetration depth and these parameters is still obvious.

[0014] The penetration depth of a rigid projectile is mainly controlled by two dimensionless physical quantities (i.e., the impact function I and the fragment shape function N), and the formula is as follows:

[0015]

[0016] Define the dimensionless number:

[0017]

[0018] The dimensionless number is crucial in the mechanics of rigid fragment penetration. The impact function and the fragment shape function are respectively determined by the mass, velocity, yield strength, and thickness of the target, and affect the penetration depth. The larger the impact function, the greater the penetration depth; while the fragment shape function is related to the geometric shape and size of the fragment and also significantly affects the penetration effect.

[0019] Obviously, the dimensionless number I / N is similar to the combined parameter in the Forrestal formula, and in the combined parameter, V1 considers the interaction force with the target during the penetration process in addition to the initial velocity V.

[0020] On this basis, in order to improve the model accuracy and prediction ability, the present invention can incorporate the combined parameter Z as an additional input into the model. This combined parameter reflects the comprehensive influence of the target material characteristics and fragment performance, and has a significant relationship with the penetration depth, which can simplify the calculation and provide stronger physical background support.

[0021] (1.2) Experimental data preprocessing and feature fusion

[0022] (1.2.1) Original data standardization: Normalize the experimental data concrete dataset to eliminate the dimension difference. Use the standardization formula where x is the original data value, μ is the mean of the data, σ is the standard deviation of the data, and z is the standardized data. Ensure that features such as mass, velocity, target material strength, etc. are in the same dimension space, so that the data of each attribute has the same scale, thereby improving the stability and prediction accuracy of model training.

[0023] (1.2.2) Physical feature enhancement: Incorporate the combined parameter Z as a new feature and fuse it with the original experimental data to construct an input feature set X = [v, m, d, f c , CRH, l n , ρ t , Z], forming an input matrix that takes into account both data distribution and physical laws.

[0024] (1.3) Input adaptation of physical features and neural network: Use a fully connected layer as the input layer to map the fused feature set X to a high-dimensional space. The number of neurons in the input layer is consistent with the feature dimension to ensure the lossless transmission of physical features.

[0025] (2) Design a multi-objective loss function. Combine the data fitting loss, physical law loss, and boundary loss. The data fitting loss constrains the neural network training, the physical law loss ensures that the model output conforms to the positive correlation between the penetration depth and Z, and the boundary loss constrains the predicted value of the penetration depth to be non-negative, which conforms to the physical boundary conditions. And automatically balance the optimization weights of physical laws and data fitting during the training process, taking into account the optimization of physical laws and model performance.

[0026] (2.1) Design of data fitting loss term

[0027] Use the mean square error (MSE) as the data fitting loss term to calculate the difference between the model prediction value and the experimental value:

[0028]

[0029] where y i is the experimental value, is the model prediction value, and N is the number of samples. During the iterative training process of the neural network, by minimizing L data, constrain the statistical consistency between the model output and experimental data to ensure the fitting accuracy of the model in the data distribution.

[0030] (2.2) Design of the physical constraint loss term

[0031] Based on the above physical analysis using the Forrestal formula, the positive correlation between the penetration depth and the combined parameter Z is obtained. The physical law is verified by slightly perturbing the input features:

[0032] Assume that the input feature X is the input of the model, and the output P is the predicted penetration depth. This method hopes that the penetration depth conforms to the physical law as the input feature increases, ensuring that the model prediction result is consistent with the known physical relationship.

[0033] Design steps:

[0034] Step 1: Obtain the original prediction: For each input X = x, the model generates a preliminary prediction value P(x) according to the current parameters:

[0035] P(x) = F(x, θ)

[0036] where F represents the prediction function of the model, x is the input feature vector, and θ is the parameter of the model.

[0037] Step 2: Slightly change the input feature: Add a small perturbation ∈ to the parameter Z in the original input x, satisfying:

[0038] |ε| << x

[0039] Get the new input x' = x + ∈, where ∈ is the small increment of the input feature Z.

[0040] Step 3: Obtain the new prediction value: Using the new input x' = x + ∈, the model generates a new prediction value P'(x'):

[0041] P'(x') = F(x + ε, θ)

[0042] Step 4: Calculate the physical loss: By comparing the original prediction P(x) and the perturbed prediction P'(x'), the present invention evaluates whether there is a violation of the physical law. At this time, the calculation of the physical loss depends on the following rules:

[0043]

[0044] If P'(x') ≥ P(x), that is, the output follows the physical law, the loss is 0. If P'(x') < P(x), that is, the output violates the physical law, the loss is P(x) - P'(x'), punishing the prediction that violates the physical law

[0045] (2.3) Design of the boundary constraint loss term

[0046] The boundary condition requires that the penetration depth K pred be greater than zero. During the training process, by penalizing those cases where the predicted value is less than zero, it is ensured that the output of the model conforms to the physical reality. The boundary loss function is as follows:

[0047]

[0048] This loss function ensures that during the training process, the output penetration depth of the model is always greater than zero, thus conforming to the physical boundary conditions.

[0049] (2.4) Dynamic weight adjustment mechanism

[0050] In order to comprehensively consider data fitting, physical constraints, and boundary conditions, the present invention combines the above loss functions into a total loss function. The total loss function L total can be expressed as:

[0051] L total = L data + λ1L physics + λ2L boundary

[0052] where L data is the standard data fitting loss, usually the mean square error (MSE) or others; L physics is the physical constraint loss; L boundary is the boundary loss; λ1, λ2 are hyperparameters used to balance each loss.

[0053] In machine learning modeling, the design of the loss function needs to take into account both the data fitting accuracy and the physical law constraints. Reasonably balancing the weights of each loss term is crucial: if the weight of the data fitting loss is too large, the model is prone to overfitting and ignores the physical laws; if the weight of the physical constraint loss is too high, the data fitting ability may be sacrificed. Therefore, a dynamic weight adjustment strategy is adopted:

[0054] For the data fitting loss (L data ) and the physical constraint loss (L physics ), an adaptive weight λ1 is set

[0055]

[0056] where ε = 10 -6 is used for numerical stability. This design ensures that when the data fitting error is large, the model focuses on optimizing the fitting accuracy; when the deviation from the physical law is significant, the physical constraints are strengthened.

[0057] The weight λ2 of the boundary loss (L boundary ) is set to a fixed value r2

[0058] λ2 = r2

[0059] Because the correspondence must strictly meet the physical boundary conditions. Fixed weights ensure that the boundary constraints are always in effect and are not affected by fluctuations in other loss terms. This dynamic balance mechanism enables the model to simultaneously satisfy the accurate fitting of experimental data, the consistency of physical laws, and the hard constraints of boundary conditions.

[0060] (3) Model training and prediction. Using the physical feature set containing the combined parameter Z as the input, a multi-objective loss joint optimization is carried out using the backpropagation neural network framework, and the network parameters are iteratively updated through the gradient descent algorithm. Finally, the predicted value of the penetration depth that satisfies the dual constraints of physical laws and data distribution is output.

[0061] Compared with the prior art, the advantages of the present invention are as follows:

[0062] 1. Deep integration of physical mechanism and data-driven, significantly improving the prediction accuracy of small samples

[0063] By analyzing the physical laws of the Forrestal formula, the combined parameter Z is constructed as the key feature input, and the energy balance relationship of the projectile-target interaction is explicitly encoded into the model. This design effectively solves the overfitting problem caused by scarce data in the prediction of explosion damage effects by traditional data-driven methods.

[0064] 2. Design of a multi-objective dynamic loss function to achieve an adaptive balance between physical laws and data distribution

[0065] A composite loss function including a data fitting term L data , a physical constraint term L physics , and a boundary constraint term L boundary is proposed, and a weight dynamic adjustment mechanism is innovatively introduced. This strategy focuses on data fitting in the initial stage of training for rapid convergence, and strengthens physical constraints in the later stage to ensure that the prediction conforms to the theory, solving the suboptimal solution problem in the data-physics trade-off of traditional models. Description of the Drawings

[0066] Figure 1 is the flowchart of the method of the present invention;

[0067] Figure 2 is the neural network structure diagram.

[0068] Figure 3 is the graph of the comparative experiment results. Detailed Embodiments

[0069] The present invention will be further described in detail below with reference to the drawings.

[0070] As Figure 1As shown in the figure, the present invention first analyzes the mechanism of projectile-target interaction based on the Forrestal formula, constructs the combined parameter Z as the key physical feature, and fuses it with the standardized test data to form the enhanced input feature set x. A multi-objective loss function is designed, including data fitting loss, physical law loss, and boundary loss, and adaptive optimization is achieved through a dynamic weight adjustment strategy. As Figure 2 shown, the invention adopts a backpropagation neural network architecture. For the original input feature set x, a small perturbation ∈ is added to the parameter Z in the original input x to form a new input x'. The fully connected layer realizes the spatial mapping of features by adjusting the weight matrix to ensure that the physical characteristics of the input data are retained. The model output layer receives the information processed through multiple levels, generates the prediction results P' and P, and constructs a composite loss function L total = L data + λ1L physics + λ2L boundary . During the iterative optimization process, the gradient feedback of the loss function dynamically adjusts the network weights, enabling the model to strengthen the embedding of physical laws while reducing data errors. Finally, through multiple rounds of parameter updates, the collaborative improvement of prediction accuracy and physical consistency is achieved, thus completing the end-to-end mapping from the original input to the physically compliant output.

[0071] The specific process is as follows:

[0072] (1) Construct the input by fusing physical prior knowledge

[0073] (1.1) Extract features by analyzing physical formulas: Based on the Forrestal formula

[0074]

[0075] where P represents the penetration depth, and the other input parameters include: d represents the diameter of the fragment, M represents the mass of the fragment, V represents the velocity of the fragment, f represents the compressive strength of the concrete target, pt represents the density of the concrete target, N represents the fragment shape factor (dimensionless), and S represents the dimensionless empirical constant. By fusing these physical quantities and determining the core influencing factors, the model can accurately predict the penetration depth of the fragment during the explosion under different conditions, thereby providing theoretical support for the study of explosion damage effects.

[0076] (1.1.2) Construction of combined parameters: According to the analytical solution of Forrestal, there is a certain relationship between the penetration depth (P) and the physical parameters. Specifically, the penetration depth is related to the target density (p), the fragment shape factor (N), the square of the fragment velocity (V1 2 ) and factors such as the dimensionless empirical constant (S) and the target compressive strength (f). Define the dimensionless combined parameter It is used as a key physical feature to characterize the interaction strength between the projectile and the target. By analyzing the formula, it can be found that the penetration depth has a positive correlation with Z. As Z increases, the penetration depth shows an increasing trend. In addition, although this relationship is not linear but shows asymptotic changes through logarithmic terms, in theoretical analysis, the positive correlation between the penetration depth and these parameters is still obvious.

[0077] (1.2) Data preprocessing: Standardize the original experimental data such as fragment parameters and target material properties to eliminate the dimension differences.

[0078] (1.3) Feature fusion: Combine Z with the standardized features into the input matrix X, and map it to a high-dimensional space through a fully connected layer to retain physical information.

[0079] (2) Design a multi-objective loss function

[0080] (2.1) Data fitting loss: Use the mean square error (MSE) to constrain the consistency between the model prediction value and the experimental value

[0081]

[0082] where y i is the experimental value, is the model prediction value, and N is the number of samples. During the iterative training process of the neural network, by minimizing L data , the statistical consistency between the model output and the experimental data is constrained to ensure the fitting accuracy of the model in the data distribution.

[0083] (2.2) Design of the physical constraint loss term

[0084] According to the above physical analysis based on the Forrestal formula, the positive correlation between the penetration depth and the combined parameter Z is obtained. The physical law is verified by slightly perturbing the input features:

[0085] Suppose the input feature X is the input of the model, and the output P is the predicted penetration depth. This method hopes that the penetration depth conforms to the physical law as the input feature increases, ensuring that the model prediction result is consistent with the known physical relationship.

[0086] Step 1: Obtain the original prediction: For each input X = x, the model generates a preliminary prediction value P(x) according to the current parameters:

[0087] P(x) = F(x, θ)

[0088] where F represents the prediction function of the model, x is the input feature vector, and θ is the parameter of the model.

[0089] Step 2: Slightly vary the input features: Add a small perturbation E to the parameter Z within the original input x, satisfying:

[0090] |ε| << x

[0091] Obtain the new input x′ = x + ∈, where ∈ is the small increment of the input feature Z.

[0092] Step 3: Obtain the new predicted value: Use the new input x′ = x + ∈, and the model generates a new predicted value P′(x′);

[0093] P′(x′) = F(x + ε, θ)

[0094] Step 4: Calculate the physical loss: By comparing the original prediction P(x) and the perturbed prediction P′(x′), the present invention evaluates whether there is a violation of physical laws. At this time, the calculation of the physical loss (Lphysics) depends on the following rules:

[0095]

[0096] If P′(x′) ≥ P(x), that is, the output follows the physical laws, the loss is 0. If P′(x′) < P(x), that is, the output violates the physical laws, the loss is P(x) - P′(x′), punishing the prediction that violates the physical laws [[ID=URL]]

[0097] (2.3) Design of the boundary constraint loss term

[0098] The boundary condition requires that the penetration depth P pred be greater than zero. During the training process, by punishing those cases where the predicted value is less than zero, it is ensured that the output of the model conforms to the physical reality. The boundary loss function is as follows:

[0099]

[0100] This loss function ensures that during the training process, the penetration depth of the model output is always greater than zero, thus conforming to the physical boundary conditions.

[0101] (2.4) Dynamic weight adjustment mechanism

[0102] In order to comprehensively consider data fitting, physical constraints, and boundary conditions, the present invention combines the above loss functions into a total loss function. The total loss function L total can be expressed as:

[0103] L total = L data + λ1L physics + λ2L boundary

[0104] where L datais the standard data fitting loss, usually the mean squared error (MSE) or others; L physics is the physical constraint loss; L boundary is the boundary loss; λ1, λ2 are hyperparameters used to balance the losses of each item.

[0105] In machine learning modeling, the design of the loss function needs to take into account both the data fitting accuracy and the physical law constraints. It is crucial to reasonably balance the weights of each loss term: if the weight of the data fitting loss is too large, the model is prone to overfitting and ignores the physical laws; if the weight of the physical constraint loss is too high, the data fitting ability may be sacrificed. Therefore, a dynamic weight adjustment strategy is adopted:

[0106] For the data fitting loss (L data ) and the physical constraint loss (L<� physics ), set the adaptive weight λ1

[0107]

[0108] where ε = 10 -6 for numerical stability. This design ensures that when the data fitting error is large, the model focuses on optimizing the fitting accuracy; when the physical law deviates significantly, the physical constraint is strengthened.

[0109] The weight λ2 of the boundary loss (L boundary ) is set to a fixed value r2

[0110] λ2 = r2

[0111] because it corresponds to the physical boundary conditions that must be strictly satisfied. The fixed weight ensures that the boundary constraint is always in effect and is not affected by the fluctuations of other loss terms. This dynamic balance mechanism enables the model to simultaneously meet the accurate fitting of experimental data, the consistency of physical laws, and the hard constraints of boundary conditions.

[0112] (3) Model training and prediction

[0113] (3.1) Using the fused feature set X and the perturbed X` as inputs, jointly optimize the multi-objective loss using the backpropagation framework, and update the network parameters iteratively through gradient descent.

[0114] (3.2) The adaptive weight mechanism balances data fitting and physical constraints: when the data error is large, it focuses on L data , when the physical deviation is significant, it strengthens L physics , and the boundary loss L boundary always maintains a high priority. After training reaches the preset number of times, output the penetration depth prediction value that satisfies both the physical law and the data distribution constraints.

[0115] (3.3) After the training is completed, the model outputs the predicted value of the penetration depth that simultaneously satisfies the experimental data fitting, physical laws, and boundary conditions.

[0116] Example:

[0117] To verify the effectiveness of the present invention, experimental samples of concrete penetration depth were collected from Internet technology literature. Each piece of data contains 9 fields, and simulation data was generated through simulation calculations at the same time. First, the experimental data was divided into an experimental data training set and an experimental data test set according to a ratio of 8:2, where the experimental data test set was used to test the performance of each model. The present invention processes the experimental data to calculate a new input value Z. This new parameter synthesizes physical characteristics, and there is a positive correlation between the penetration depth and Z. By adding Z as a new input feature to the neural network model, the network can capture the interaction between the target material and the fragment characteristics, thereby improving the prediction accuracy. In this experiment, the accidental explosion concrete penetration depth prediction model integrating empirical algorithm knowledge was compared with the traditional machine learning prediction model;

[0118] 1. Data preparation

[0119] The dataset information is shown in Table 1.

[0120] Table 1 Concrete Penetration Data Table

[0121]

[0122] 2. Comparative experiment

[0123] (1) Comparison between the accidental explosion concrete penetration depth prediction integrating empirical algorithm knowledge and the traditional prediction learning model

[0124] Penetration depth prediction is a typical multi-factor coupled nonlinear physics problem. Different machine learning models exhibit distinct modeling advantages due to their inherent mechanics. Support vector machines (SVMs), based on the principle of structural risk minimization, use kernel techniques to construct an optimal hyperplane in feature space. Their characteristic of maximizing classification margins enables them to maintain strong generalization capabilities even with small sample sizes. They are particularly adept at handling high-dimensional nonlinear mappings between fragment parameters and target media. Convolutional neural networks (CNNs), with their local connections and weight sharing mechanisms, effectively extract the spatial gradient features of projectile-target interactions during penetration. They capture the spatial correlation between material deformation and energy dissipation through the cascade of multiple convolutional kernels. XGBoost, an optimized implementation of gradient boosting decision trees, uses a second-order Taylor expansion for loss function optimization and regularization strategies to control model complexity. It demonstrates excellent predictive stability in modeling penetration dynamics parameters with significant feature interactions. LSBoost (least squares boosting) iteratively optimizes a weighted combination of base learners, progressively correcting residuals using the least squares criterion. It is particularly well-suited for handling heteroscedasticity between material parameters and penetration depth. This experiment uses the mean square error (MSE) to build an evaluation system: MSE characterizes the absolute magnitude of the prediction error through the square loss function, which is particularly suitable for engineering scenarios where the penetration depth varies across orders of magnitude. The experimental results are shown in Table 2 and Figure 3 shown.

[0125] Table 2 Comparison of concrete penetration prediction results

[0126]

[0127] Specifically, in concrete penetration experiments, the proposed model achieved a lower mean square error (MSE), indicating a reduced prediction error and more accurate results. This indicates that the introduction of physical prior knowledge significantly reduces the model's prediction error.

[0128] The proposed model adheres to physical laws and constraints during the learning process, reducing the model's degrees of freedom and the potential for overfitting. Physical knowledge effectively constrains the model's prediction space, enabling the backpropagation neural network to better align with real-world physical phenomena when solving complex physical problems, thereby improving the model's stability and generalization capabilities.

Claims

1. An unexpected explosion concrete penetration depth prediction method integrating empirical algorithm knowledge, characterized in that, The method includes the following steps: (1) Construct the input by integrating physical prior knowledge. Analyze the physical relationship between the penetration depth and the fragment velocity, density, shape factor, and the compressive strength of the target based on the Forrestal formula, and derive the combined parameter Z as an additional input feature of the model. Integrate it with the experimental data and input it into the neural network. (2) Design a multi-objective loss function. Combine the data fitting loss, physical law loss, and boundary loss. The data fitting loss constrains the neural network training. The physical law loss ensures that the model output conforms to the positive correlation between the penetration depth and Z. The boundary loss constrains the predicted penetration depth to be non-negative, which conforms to the physical boundary conditions. And automatically balance the optimization weights of the physical law and data fitting during the training process, taking into account the optimization of both the physical law and model performance. (3) Model training and prediction. Use the physical feature set containing the combined parameter Z as the input, adopt the backpropagation neural network framework to jointly optimize the multi-objective loss, and iteratively update the network parameters through the gradient descent algorithm. Finally, output the predicted penetration depth that satisfies the dual constraints of physical laws and data distribution.

2. According to claim (1), construct the input by integrating physical prior knowledge. In this step, by analyzing the physical mechanism of the Forrestal formula, generate the combined parameter Z, and combine technologies such as experimental data standardization and feature fusion to construct an input feature set with clear physical meaning. This design not only retains the flexibility of data-driven methods but also constrains the model hypothesis space through physical parameters, solving the problems of overfitting and prediction distortion prone to occur in traditional methods when data is scarce. The specific steps of step (1) are as follows: (1.1) Physical feature analysis and combined parameter generation based on the Forrestal formula. Among them, Let \(P\) represent the penetration depth (unit: m), and the other input parameters include: \(d\) represents the diameter of the fragment (unit: m), \(M\) represents the mass of the fragment (unit: kg), \(V\) represents the velocity of the fragment (unit: m / s), \(f\) represents the compressive strength of the concrete target (unit: Pa), \(\rho_t\) represents the density of the concrete target (unit: kg / m 3 ), \(N^*\) represents the fragment shape factor (dimensionless), and \(S\) represents a dimensionless empirical constant. By integrating these physical quantities to determine the core influencing factors, the model can accurately predict the penetration depth of the fragment during the explosion under different conditions, thus providing theoretical support for the study of explosion damage effects. (1.1.2) Composite parameter construction. According to Forrestal's analytical solution, there is a certain relationship between the penetration depth (P) and physical parameters. Specifically, the penetration depth is related to factors such as the target density (p), the fragment shape factor (N), the square of the fragment velocity (V1 2 ), as well as the dimensionless empirical constant (S) and the target compressive strength (f). Define the dimensionless composite parameter and use it as the key physical feature characterizing the interaction strength between the projectile and the target. By analyzing the formula, it can be found that the penetration depth has a positive correlation with Z. This parameter comprehensively quantifies the dynamic balance relationship between kinetic energy and the anti-penetration ability of the target material, providing physical prior guidance for the model. Specifically, the change in the penetration depth P obtained from the Forrestal formula is affected by Z. As Z increases, the penetration depth shows an increasing trend. In addition, although this relationship is not linear but shows asymptotic change through logarithmic terms, in theoretical analysis, the positive correlation between the penetration depth and these parameters is still obvious. (1.1.1) First, analyze the physical formula. According to the analytical solution of the Forrestal formula: The penetration depth of a rigid projectile is mainly controlled by two dimensionless physical quantities (i.e., the impact function I and the fragment shape function N). The formula is as follows: Define the dimensionless number: The dimensionless number is crucial in the mechanics of rigid fragment penetration. The impact function and the fragment shape function are respectively determined by the mass, velocity of the fragment, the yield strength and thickness of the target, and affect the penetration depth. The larger the impact function, the greater the penetration depth. The fragment shape function is related to the geometric shape and size of the fragment and also significantly affects the penetration effect. Obviously, the dimensionless number I / N is similar to the combined parameter in the Forrestal formula. In the combined parameter, V1 considers not only the initial velocity V but also the interaction force with the target during the penetration process. On this basis, in order to improve the model accuracy and prediction ability, the present invention can incorporate the combined parameter Z as an additional input into the model. This combined parameter reflects the comprehensive influence of the target material properties and fragment performance, shows a significant relationship with the penetration depth, and can simplify the calculation and provide stronger physical background support. (1.2.1) Original data standardization: Normalize the experimental data concrete dataset to eliminate the dimensional differences. Use the standardization formula where x is the original data value, μ is the mean of the data, σ is the standard deviation of the data, and z is the standardized data. Ensure that features such as quality, speed, and target strength are in the same dimensional space, making the data of each attribute have the same scale, thereby improving the stability and prediction accuracy of model training. (1.2.2) Physical Feature Enhancement: The combined parameter Z is used as a new feature to fuse with the original experimental data, and the input feature set X = [v, m, d, f c , CRH, l n , ρ t , Z] is formed to create an input matrix that takes into account both data distribution and physical laws. (1.2) Preprocessing of experimental data and feature fusion (1.3) Input adaptation of physical features and neural network: Use a fully connected layer as the input layer to map the fused feature set X to a high-dimensional space. The number of neurons in the input layer is consistent with the feature dimension to ensure the lossless transmission of physical features.

3. Design a multi-objective loss function according to claim (2). In this step, by designing three loss functions for data fitting, physical laws, and boundary constraints, and combining a dynamic weight adjustment mechanism, an adaptive balance between data-driven learning and physical prior knowledge of the model is achieved. The specific steps for collaborative optimization of the multi-objective loss function are as follows: (2.1) Design of the data fitting loss term The mean square error (MSE) is used as the data fitting loss term to calculate the difference between the model prediction value and the experimental value: where y i is the experimental value, is the model predicted value, and N is the number of samples. During the neural network iterative training process, by minimizing L data , the statistical consistency between the model output and the experimental data is constrained to ensure the fitting accuracy of the model in the data distribution. (2.2) Design of the physical constraint loss term According to the above physical analysis based on the Forrestal formula, the positive correlation between the penetration depth and the combined parameter Z is obtained, and the physical law is verified by slightly perturbing the input features: Assume that the input feature X is the input of the model, and the output P is the predicted penetration depth. This method hopes that the penetration depth conforms to the physical law as the input feature increases, ensuring that the model prediction result is consistent with the known physical relationship. Design steps: Step 1: Obtain the original prediction: For each input X = x, the model generates a preliminary prediction value P(x) according to the current parameters: P(x) = F(x, θ) Among them, F represents the prediction function of the model, x is the input feature vector, and θ is the parameter of the model. Step 2: Slightly change the input feature: Add a small perturbation ∈ to the parameter Z in the original input x, satisfying: |ε| << x Get the new input x' = x + ε, where ε is the small increment of the input feature Z. Step 3: Obtain the new prediction value: Use the new input x' = x + ε, and the model generates a new prediction value P'(x'): P'(x') = F(x + ε, θ) Step 4: Calculate the physical loss: By comparing the original prediction P(x) and the perturbed prediction P ′ (x ′ ), the present invention evaluates whether there is a violation of physical laws. At this time, the calculation of the physical loss depends on the following rules: If P'(x') ≥ P(x), that is, the output follows the physical law, the loss is 0. If P'(x') < P(x), that is, the output violates the physical law, the loss is P(x) - P'(x'), and the prediction that violates the physical law is punished (2.3) Design of the boundary constraint loss term The boundary condition requires that the penetration depth P pred be greater than zero. During training, by penalizing those cases where the predicted values are less than zero, it is ensured that the output of the model conforms to physical reality. The boundary loss function is as follows: This loss function ensures that during the training process, the output penetration depth of the model is always greater than zero, thus conforming to the physical boundary conditions. (2.4) Dynamic weight adjustment mechanism To comprehensively consider data fitting, physical constraints, and boundary conditions, the present invention combines the above loss functions into a total loss function. The total loss function L total can be expressed as: L total = L data + λ1L physics + λ2L boundary where L data is the standard data fitting loss, usually the mean squared error (MSE) or others; L physics is the physical constraint loss; L boundary is the boundary loss; λ1, λ2 are hyperparameters used to balance each loss. In machine learning modeling, the design of the loss function needs to consider both the data fitting accuracy and the physical law constraints. It is crucial to reasonably balance the weights of each loss term: if the weight of the data fitting loss is too large, the model is prone to overfitting and ignores the physical law; if the weight of the physical constraint loss is too high, the data fitting ability may be sacrificed. Therefore, a dynamic weight adjustment strategy is adopted: For the data fitting loss (L data ) and the physical constraint loss (L physics ), an adaptive weight λ1 is set where ε = 10 -6 for numerical stability. This design ensures that when the data fitting error is large, the model focuses on optimizing the fitting accuracy; when the deviation of physical laws is significant, the physical constraints are strengthened. The weight λ2 of the boundary loss (L boundary ) is set to a fixed value r2 λ2 = r2 Because it corresponds to the physical boundary conditions that must be strictly satisfied. The fixed weight can ensure that the boundary constraint is always in effect and is not affected by the fluctuations of other loss terms. This dynamic balance mechanism enables the model to simultaneously meet the precise fitting of experimental data, the consistency of physical laws, and the hard constraints of boundary conditions.

4. According to the model training and prediction in claim (3), in the present invention, a backpropagation neural network is selected as the model for predicting the penetration depth. Since the relationship between the penetration depth and multiple physical parameters is very complex and non-linear, the backpropagation neural network can capture these complex interaction relationships through its multi-layer structure. During the network training process, through an iterative optimization algorithm based on gradient descent and a backpropagation mechanism, the system minimizes a composite loss function that includes a data fitting term, a physical constraint term, and a boundary condition term, thereby ensuring that the prediction results not only satisfy the statistical consistency of the experimental data but also strictly follow the physical law constraints of the Forrestal formula and conform to the basic boundary condition of non-negative penetration depth.

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