Method for constructing artillery bore opening flow field proxy model based on physical information neural network

Abstract

The invention discloses an artillery bore opening flow field proxy model construction method based on a physical information neural network, and belongs to the technical field of test evaluation. According to the artillery bore opening flow field proxy model construction method based on the physical information neural network, N-S solution loss and empirical formula loss in a loss function are combined through the hard constraint thought, and the loss function is simplified. The method firstly describes a basic framework of the method, secondly introduces a specific formula of an improved loss function, and finally verifies the superiority of the method through experiments. According to the method, the shock wave field reconstruction has obvious advantages when the data volume is low, response values of all design points in the space can be quickly obtained, and the optimization efficiency is greatly improved.

Description

Technical Field

[0001] The invention belongs to the technical field of test evaluation. Background Art

[0002] The gun muzzle shock wave field test is one of the key assessment indicators of the service performance and safety performance of the gun design finalization test. During the shock wave field test, it is impossible to accurately and comprehensively obtain the overpressure distribution law of the shock wave field due to the concentration of points in a specific area. Therefore, how to reconstruct the shock wave field pressure distribution under the condition of limited measurement points in a small range is a hot issue in the field of shock wave field research.

[0003] At present, there are two main methods for reconstructing the shock wave overpressure field under limited measurement points: the first is a shock wave reconstruction algorithm based on statistical data, including B-spline interpolation method, cubic spline interpolation algorithm, biharmonic spline surface interpolation algorithm and radial basis function network interpolation algorithm based on the peak overpressure of the measurement point. These methods use fixed statistics or assume that the data follows a specific distribution to fill in the missing values, ignoring the physical laws of shock wave propagation, resulting in significant errors in data reconstruction and having great limitations; the second is a data-driven neural network represented by back propagation (BP). The main limitation of this method is that the weight distribution of the test area is too high and the non-test area lacks training data, resulting in overfitting in the test area and underfitting in the non-test area. If the physical propagation law of shock waves and sparse data characteristics can be combined, it will be one of the means to accurately and comprehensively reconstruct the shock wave field under limited measurement points.

[0004] The Physical Information Neural Network (PINN) is a new network architecture that combines physical laws with deep learning technology. It uses physical laws as prior knowledge for network training to improve the prediction accuracy of the model. The proxy model constructed by this network further reduces the training samples while obtaining high-precision reconstruction results by means of data dimensionality reduction. This method has been applied in the reconstruction of sparse data physical fields in multiple scenarios: Lu Zhibin et al. improved the physical information neural network proxy model and reconstructed the two-dimensional temperature field between plates with 10 samples, with a maximum absolute error of only 0.0038℃; Luo Jinwen et al. constructed a deep neural network proxy model to reconstruct the pressure field of the equipment in water, using 57 sample points as training data, with the maximum error and root mean square error less than 1%; Huang Li et al. input 50 sample points into the neural network proxy model to reconstruct the temperature field of the power cable, with a root mean square error of 0.00398℃. In summary, the neural network proxy model has a low error when reconstructing the physical field with low data volume, and the two papers use the NS equations as physical information to construct the network, while the shock wave field propagation process is a gas flow process, which can also be described by the NS equations. Summary of the invention

[0005] The purpose of the present invention is to combine the NS solution loss and the empirical formula loss in the loss function by adopting the hard constraint idea, so as to achieve a method for constructing a gun muzzle flow field proxy model based on a physical information neural network with a simplified loss function.

[0006] The steps of the present invention are: S1. The Henrych overpressure formula and the defense design specification formula are combined and incorporated into the loss function as additional constraints. The specific formula is as follows: Where Δp is the peak overpressure, R is the distance from the test node to the explosion center, w is the TNT equivalent of the warhead; the loss function is specifically constructed as follows: Weight W 1 With W 2 Represents the residual MSE of neural network data 1 The mean square error (MSE) between the empirical formula and the true value 2 The weight of .

[0007] The present invention has significant advantages in reconstructing the shock wave field when the amount of data is low, and the response values ​​of all design points in the space can be quickly obtained, thereby greatly improving the efficiency of optimization. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 It is a physical neural network framework diagram; Figure 2 It is a visualization of the shock wave field; Figure 3 It is a visualization diagram of the PINN and BP reconstruction errors under ideal conditions; (a): improved PINN prediction shock wave field; (b): improved PINN prediction error field; (c): BP prediction shock wave field; (d): BP prediction error field; Figure 4 is the training data and error curve. DETAILED DESCRIPTION

[0009] The invention takes the proxy model of physical information neural network as the entry point, takes the finite element solution result of NS equation as the physical information of the network and reduces the dimension to obtain the original data set; takes the shock wave overpressure empirical formula as an additional constraint into the loss function to prevent the network training from overfitting. The invention first describes the basic framework of the method, then introduces the specific formula of the improved loss function, and finally verifies the superiority of the method through experiments.

[0010] Proxy model building process For a given design space, the surrogate model can be used as a substitute for the high-precision numerical model. Its advantage is that the response values ​​of all design points in the space can be quickly obtained, thereby greatly improving the efficiency of optimization. This paper sorts out the construction process of the surrogate model.

[0011] The simplified design process can be divided into the following three steps (1) Use appropriate methods to reduce the dimension of the optimization problem and reduce the difficulty of optimization; (2) extracting initial samples from the design space after dimensionality reduction to obtain an initial sample space; (3) Use the reduced-dimensional sample data to input the neural network to obtain the initial proxy model.

[0012] When drawing random samples, random sampling is to select sample points completely randomly from the design space. d For a set of n sample points in space, the random sampling method can be expressed as

[0013] A major problem with this approach is that the sample points may be clustered in certain areas within the design space, resulting in uneven coverage. Therefore, although the entire design space may be uniformly covered in a statistical sense, in the case of finite samples, random sampling does not provide ideal coverage in the space, and is prone to uncovered areas and local overcrowding.

[0014] The optimal Latin hypercube sampling method ensures that the sample points are evenly distributed in each dimension by dividing the space, and on this basis maximizes the minimum distance between all sample points. In d-dimensional space, if x i and x j are two sample points whose Euclidean distance satisfies

[0015] The initial samples extracted by this method can better express the characteristics of the entire space. In the experiment, the optimal Latin hypercube method is used to sample the original shock wave field peak pressure data to obtain the original data set for proxy model training.

[0016] Governing equations Partial differential equations can be used to describe physical phenomena. For the problem of shock wave field propagation, the peak pressure variation with space is usually used as the solution of the partial differential equation. The control equation of shock wave field propagation adopts the two-dimensional axisymmetric Reynolds averaged NS equation: Where: Where: ρ, p, u, v, e, Re are density, pressure, average velocity in x and y directions, total energy per unit volume, and Reynolds number, respectively. Total energy per unit volume γ is the gas specific heat ratio; ρ 0 ,ω 0 , μ 0 are all parameters at the center of the jet inlet; D is the characteristic length, and for the muzzle flow field calculation, the gun caliber value is taken. The coefficient δ determines the flow type, and δ = 1 is a two-dimensional axisymmetric flow.

[0017] Physical information neural network framework and its loss function improvement As a kind of proxy model, neural network has the characteristics of strong nonlinear mapping ability, flexible model and high accuracy. In recent years, Physical Information Neural Network (PINN) uses neural network to approximate the solution of physical problems. This method embeds physical principles (in the form of partial differential equations) as prior knowledge into the model, and integrates the solution constraints as restrictive factors into the loss function. By penalizing the residual of partial differential equations, it ensures that the results predicted by the network are consistent with the real physical laws. Its specific framework is as follows Figure 1 shown.

[0018] Figure 1 In the equation, NN represents neural network, PDE represents the two-dimensional axisymmetric Reynolds averaged NS equation describing the propagation of shock wave field, ε represents the set threshold, σ represents the neural network weight parameter, X represents the horizontal coordinate of the test point, Y represents the vertical coordinate of the test point, P represents the peak pressure of the test point, u represents the peak pressure predicted by the network, and P r To solve the NS equation, we get the true value. If the loss function converges to the threshold, we output it. If it does not reach the threshold, we continue to return to the forward network for training.

[0019] The shock wave overpressure empirical formula can better fit the relationship between the distance from the test point to the muzzle and the pressure when the gun is fired. This paper combines the Henrych overpressure formula and the defense design specification formula into the loss function as an additional constraint condition. The specific formula is as follows: Where, Δp is the peak overpressure, in MPa; R is the distance from the test node to the explosion center, in meters; w is the TNT equivalent of the warhead, in kg.

[0020] The specific structure of the loss function is as follows:

[0021] When designing the loss function, we consider the true value P r , network prediction value P p And the approximate value P calculated by the empirical formula l In this framework, the weight W 1 With W 2 Represents the residual MSE of neural network data 1 The mean square error (MSE) between the empirical formula and the true value 2 Ideally, if the neural network can effectively solve the partial differential equation (PDE) problem, MSE 1 will approach zero; at the same time, if the prediction of the neural network not only conforms to the solution of the PDE, but also coincides with the empirical formula, then the MSE 2 will also tend to zero. When the value of the loss function is zero, it means that the predicted value of the neural network on the training set is very close to the true value, thus transforming the problem into how to optimize the loss function. To this end, we impose a hard constraint so that W 1 +W 2 = 1, and dynamically adjust W through the adaptive weight method 1 With W 2 The application of adaptive weights improves the immediate response of the model to the error term, optimizes the allocation of training resources, thereby improving the efficiency of learning and helping to achieve better performance balance and error reduction in iterative training.

[0022] Simulation and Experiment The experiment introduces the construction of the initial data set, obtains the network training set through the most appropriate Latin hypercube sampling, explores the reconstruction error under the same parameters of BP and this method, and the relationship between the number of sample points and the reconstruction error. Experimental platform: Python is used as the development language, the experimental process is carried out under the win10 operating system, the processor model is i7-12700H, the graphics card model is RTX3060, the programming platform is pycharm, and the deep learning framework is pytorch1.8.1.

[0023] Measurement point coordinates and data acquisition The construction of the proxy model requires initial data support. This paper designs a data sample acquisition method based on finite elements. In order to make the data obtained by finite element simulation closer to the real shock wave field data, the low-resolution observation point setting in this experiment is consistent with the sensor placement position in the national military standard muzzle shock wave and other overpressure curve measurements. (0,0) is the muzzle position, and a 15×15 matrix sensor array is placed at a spacing of 0.5m on the rear side of the simulated gun body, with a total of 225 observation points. Some data are shown in Table 1. The original data visualization is shown in Table 1. Figure 2 shown.

[0024] Table 1 Overpressure peak value table of some measuring points in finite element simulation (unit: kpa)

[0025] Network specific parameter settings and evaluation indicators The optimal Latin hypercube sampling was used to downsample the observation points by 4 times, and a total of 56 points were selected as the input of PINN, and the remaining points were used as the validation set. The hyperbolic tangent (tanh) was selected as the activation function and Adam was used as the network optimizer. To suppress overfitting, both BP and PINN used a simple NLP network with the number of layers being [2, 13, 1] and the number of iterations being set to 50,000.

[0026] The evaluation indicators are mean absolute error and relative L2 norm. The mean absolute error is the true value P r and the predicted value P p The sum of the absolute values ​​of the differences represents the average error margin of the predicted value, without considering the direction of the error. The formula is as follows:

[0027] The relative L2 norm error calculation formula is: Among them, P r represents the true value, P p Represents the predicted value, the relative L2 norm error itself is a dimensionless value, which only represents a ratio of the relative difference between the prediction and the truth.

[0028] The shock wave field reconstruction experiment verifies that since the shock wave propagation is symmetrical, in order to facilitate the observation of the macroscopic reconstruction effect, the data is mirrored in a direct symmetrical manner on the negative semi-axis only when drawing. In terms of numerical error, the experiment compares the field reconstruction of the data-driven BP neural network, and the visualization effect of the experimental prediction data is compared with the real data processed by the NS equation as prior knowledge. Figure 3 . The reconstruction error under ideal conditions is shown in Table 2

[0029] Table 2 Comparison of improved PINN and BP errors under ideal conditions

[0030] according to Figure 3, the relative L2 norm error and mean absolute error of the improved PINN are slightly lower than those of the BP neural network. The large error areas are concentrated in the range of the horizontal axis [0,2] and the vertical axis [0,2]. Due to the addition of the overpressure empirical formula as a constraint, the overall error of PINN is lower than that of BP. The maximum fitting error of BP is at (1.5,0), with an error of 2.343246 kPa; the minimum fitting error is at (2,4.5), with an error of 0.02265 kPa; the maximum fitting error of PINN is at (1.5,0), with an error of 2.076952 kPa; the minimum fitting error is at (2,5.5), with an error of only 0.007307 kPa. According to Table 2, the relative L2 norm error of PINN is reduced by 25.876% compared with BP, and the mean absolute error is reduced by 9.5%.

[0031] To verify the reconstruction performance of the network under low data volume, 60 points are used as the test set. The training data starts from 10 and increases by 10 each time until 160. To avoid contingency, 50 Monte Carlo experiments are performed in each case to take the average value, and the average absolute error curves of BP and improved PINN under different training data are plotted, as shown in the figure. Figure 4 shown.

[0032] Overall, the error of improved PINN is better than that of BP, and the MAE is less than 0.56kpa in the whole interval. The reconstruction error is negatively correlated with the number of training sets. When the number of training sets is equal to 10, the error is the largest due to the small amount of data. The error decreases rapidly when the number of training sets is continuously increased until the number of training sets is 80, and then the error decreases slowly when the number of training sets is added. When the number of training data is 10, the error of the two methods is the largest, the MAE of BP is 0.6360765kpa, and the MAE of improved PINN is 0.55412259kpa, which is 12.88% lower than BP; when the number of training data is 160, the error of the two methods is the smallest, the MAE of BP is 0.0974749kpa, and the MAE of improved PINN is 0.09477833kpa, which is 2.7% lower than BP. It can be seen that this method has significant advantages in reconstructing shock wave fields when the amount of data is low.

[0033] Conclusion: In order to solve the problem of insufficient accuracy of shock wave field reconstruction under low data volume, this paper proposes a shock wave field reconstruction agent model based on physical information neural network. The NS equation is built into the network as a priori physical law, the network loss function is hard constrained, and the shock wave overpressure empirical formula is built into the loss function. Experiments show that the L2 norm error and mean absolute error of this method reach 2.367699×10 -3 and 1.385007×10 -1kpa, decreased by 25.876% and 9.5% respectively compared with BP; the reconstruction performance of this model is better than BP when the amount of data is low. When the training data is 10, the MAE decreased by 12.88% compared with BP.

Claims

1. A method for constructing a gun muzzle flow field proxy model based on a physical information neural network, characterized in that: The steps are: S1. The Henrych overpressure formula and the defense design specification formula are combined and incorporated into the loss function as additional constraints. The specific formula is as follows: Where Δp is the peak overpressure, R is the distance from the test node to the explosion center, and w is the TNT equivalent of the warhead; The specific composition of the loss function is as follows: The weights W1 and W2 represent the weights of the neural network data residual MSE1 and the mean square error MSE2 between the empirical formula and the true value, respectively.

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