GH4169 high-temperature alloy constitutive model construction method based on double-layer PSO-BP neural network

Abstract

The invention discloses a GH4169 high-temperature alloy constitutive model construction method based on a double-layer PSO-BP neural network, and belongs to the field of material science, and the method comprises the steps: obtaining stress-strain data through a Hopkinson pressure bar experiment, and constructing a sample library; executing a double-layer PSO algorithm to optimize hyper-parameters and connection parameters of the BP neural network; using the optimized parameters to train a BP neural network, and generating a construction model; the stress-strain behavior under the new working condition is predicted, and RMSE and R2 are calculated to evaluate the model precision. A BP neural network is used as a basic framework, a double-layer particle swarm optimization (PSO) is introduced to optimize parameters of the GH4169 high-temperature alloy, and a more reliable theoretical support is provided for the prediction precision of mechanical properties of the GH4169 high-temperature alloy under complex physical loading conditions and multi-stress states and engineering application of the GH4169 high-temperature alloy.

Description

Technical Field

[0001] This invention belongs to the field of materials science, specifically, it relates to a method for constructing a constitutive model of GH4169 high-temperature alloy based on a two-layer PSO-BP neural network. Background Technology

[0002] GH4169 is a precipitation-strengthened nickel-based superalloy. In practical engineering applications, this alloy is often subjected to complex loading environments with multiple coupled parameters (such as the combined effect of impact loads and temperature). Traditional theoretical models struggle to fully describe its nonlinear mechanical behavior, easily leading to prediction biases. Neural networks, with their multi-layered neuron structure, can adaptively learn the evolution of stress-strain under different working conditions. By flexibly incorporating multi-dimensional input features such as strain rate, loading rate, and ambient temperature, they significantly improve the prediction accuracy of stress-strain curves under complex conditions, providing more reliable mechanical data support for the impact-resistant design of structural components in aerospace and other fields.

[0003] Obtaining stress-strain data for GH4169 (especially under high temperature and high strain rate conditions) typically requires complex experimental methods such as the Hopkinson bar (SHPB). This process is affected by sample preparation, equipment precision, and the coupling of multiple factors, resulting in long cycles, high costs, and limited data samples. Neural networks, with their powerful nonlinear fitting capabilities, can uncover the implicit relationships between parameters such as stress, strain, strain rate, and temperature based on limited, high-quality experimental data. This enables high-precision predictions under small sample conditions, significantly reducing reliance on repeated experiments and shortening the development cycle.

[0004] With the increasing demands for accuracy and efficiency in predicting the dynamic properties of materials, methods capable of integrating multi-source experimental data and achieving efficient prediction are gaining increasing attention. For the prediction of the stress-strain curve of GH4169, neural network methods have gradually become a research hotspot. This method can form a "feedback-optimization" closed loop with experimental data: the prediction results can guide the optimization of experimental schemes (e.g., prioritizing the verification of conditions with large prediction deviations), thereby improving experimental efficiency; simultaneously, new experimental data can further train the neural network model, continuously improving its generalization ability through iteration. However, due to constraints such as network structure and high dimensionality of optimization parameters, the neural network prediction process is prone to getting trapped in local optima, requiring further improvements in algorithm design and training strategies. Summary of the Invention

[0005] The purpose of this invention is to provide a constitutive model construction method for GH4169 high-temperature alloy based on a two-layer PSO-BP neural network. The method uses a BP neural network as the basic framework and introduces a two-layer particle swarm optimization (PSO) algorithm to optimize its parameters, thereby overcoming the problem that the BP neural network is prone to getting trapped in local optima and reducing prediction errors.

[0006] To achieve the above objectives, the technical solution provided by this invention is as follows: A method for constructing a constitutive model of GH4169 high-temperature alloy based on a two-layer PSO-BP neural network includes:

[0007] Step 1: Obtain the stress-strain curve of GH4169 high-temperature alloy through Hopkinson bar compression test, and establish a BP neural network training sample database;

[0008] Hopkinson bar experiments were conducted under preset loading pressure, different high temperatures, and different high strain rates to obtain the stress-strain curves of GH4169 high-temperature alloy material, and a sample database was established for training a BP neural network.

[0009] Step 2: Construct a two-layer particle swarm optimization (PSO) framework and improve the selection method of inertia weights in the inner and outer layers;

[0010] Step 21: Two-layer PSO-BP optimization framework

[0011] A two-layer PSO-BP optimization framework is constructed, which consists of an outer PSO layer and an inner PSO layer, forming a nested loop structure.

[0012] In the outer PSO layer, each particle corresponds to a set of hyperparameter combinations, and the fitness of each particle is determined by the optimization results of the embedded inner PSO layer. The outer PSO layer updates the particle positions iteratively (adjusting the hyperparameters) and eventually searches for the hyperparameter combination that optimizes the prediction accuracy of the neural network.

[0013] The inner PSO is driven by the outer PSO. Each time the outer PSO generates a set of hyperparameters, the inner PSO is initiated for optimization. The inner particles represent the combination of connection parameters (weights and thresholds). Using the prediction error of the training set as the fitness, the optimal connection parameters under the current hyperparameter structure are obtained through iterative optimization. The optimization error is fed back to the outer PSO as the basis for hyperparameter evaluation.

[0014] Step 22: Improve the method for selecting inertia weights

[0015] Inertia weight (ω) is the core parameter of Particle Swarm Optimization (PSO). It balances global exploration and local development capabilities by adjusting the strength of particle inheritance of historical velocity: the larger the value, the stronger the particle's global search capability, which is conducive to escaping local optima; the smaller the value, the more easily the particle is guided by the group's experience, which is conducive to local fine optimization.

[0016] This invention employs an adaptive approach to the inertia weight in the particle swarm optimization algorithm. The inertia weight and the number of iterations are related to the fitness of each particle (characterized by mean squared error, MSE). The dynamic adjustment logic of the inertia weight is as follows: a larger value is used in the early stages of iteration to ensure exploration breadth, and a smaller value is used in the later stages of iteration to enhance convergence accuracy. Therefore, both the inner and outer layers ultimately obtain the minimum value obtained during the iteration process. In this invention, the inertia weight change rule is as follows:

[0017] Where, ω min and ω max The minimum and maximum inertia weight coefficients are given in advance, and are generally taken as 0.4 and 0.9 respectively. Let be the average fitness of all particles at the d-th iteration, defined as:

[0018]

[0019] For a certain particle in the d-th iteration Fitness; n is the number of samples; Let be the minimum fitness of all particles at the d-th iteration, defined as:

[0020]

[0021] During iterative optimization, some particles will find better positions, while others will be in medium or poor positions. In the next iteration, particles in better positions are more likely to find better solutions, while the performance of particles in poorer positions further weakens. As iteration continues, the better particles will gradually approach or even reach the global optimum. Based on this, in each iteration update, the inertia weights are dynamically adjusted according to the fitness values ​​of the particles in the previous iteration.

[0022] Step 3: Use the two-layer particle swarm optimization algorithm (PSO) to optimize and obtain the optimal hyperparameters and connection parameters (weights, thresholds) of the BP neural network;

[0023] Step 31: Initialize the PSO parameters for both the inner and outer layers.

[0024] In the two-layer PSO-BP optimization framework, the parameters (number of iterations, population size, learning factor, etc.) of the inner PSO and the outer PSO are independent and need to be designed separately according to the optimization objectives and task characteristics of each layer.

[0025] First, the outer PSO determines the search range of hyperparameters (hidden layer neurons, learning rate, and number of backpropagation training iterations) and generates specific hyperparameter values ​​(i.e., initial positions) for each particle.

[0026] To adapt to the strong nonlinearity of GH4169 and avoid insufficient or redundant neurons, the neuron search range is determined to be [10, 20]. Under high strain rates, stress data changes drastically, requiring a smaller learning rate η to avoid training oscillations. If η < 0.005, convergence is too slow; if η > 0.02, the optimal weight region will be skipped. Therefore, this invention limits the learning rate η to [0.005, 0.02]. The signal-to-noise ratio of high strain rate data is lower than that of quasi-static experiments, requiring sufficient iterations for stable model convergence. However, when the number of iterations exceeds 800, the predicted MSE of GH4169 no longer decreases but instead increases due to overfitting noise. To ensure model convergence, the number of iterations is limited to [300, 800]. To avoid local optima under complex problems and cover potentially high-quality hyperparameter combinations, the outer PSO population size is set to 20, and all particles are assigned a set of hyperparameter combinations within the aforementioned hyperparameter range.

[0027] Secondly, the inner PSO generates connection parameters for each outer particle's hyperparameters, with the connection parameter range set to the conventional [-0.5, 0.5]. The training set MSE is calculated through forward propagation of the BP network and used as the initial fitness of the inner particles, thereby updating the individual optimality and global optimality of the inner particles.

[0028] Step 32: Calculate the fitness of each particle in the inner and outer layers and dynamically adjust the inertia weights.

[0029] In BP neural network optimization, particle positions represent the network hyperparameters optimized by the outer PSO layer and the connection parameters optimized by the inner PSO layer, respectively. The fitness function is used to quantify the network prediction accuracy. This invention uses the reciprocal of the mean squared error (MSE) as the fitness metric: the inner PSO layer optimizes the connection parameters under the corresponding hyperparameters to obtain the network's MSE on the training set; the outer PSO layer quantifies the particle fitness based on the network prediction MSE corresponding to the current hyperparameter combination. The specific calculation formulas are as follows:

[0030]

[0031] Among them, Fitness out (x) represents the fitness of the outer particles; MSE(x) represents the mean square error of the outer particles after optimization; Fitness in (ω) represents the fitness of the inner particles; MSE(ω) represents the mean square error of the inner particles after optimization.

[0032] During each iteration, the inertia weights are dynamically adjusted in the next iteration based on the fitness values ​​of the particles from the previous iteration. The core of the two-layer framework is the nested logic of "outer layer optimizing hyperparameters → inner layer optimizing connection parameters," but the iteration processes of the two PSO layers are completely independent, each possessing its own algorithm parameters. Therefore, when optimizing the neural network, the inertia weights of the inner and outer PSO layers are dynamically adjusted separately.

[0033] Step 33: By updating the velocity and position of the particle swarm, and after continuous iteration, output the optimal fitness of the swarm. The specific steps are as follows:

[0034] Step 331: Calculate the initial fitness. For each particle, decode the position into weights and thresholds for the BP neural network. Initialize the BP neural network and calculate the initial fitness value (MSE) for each particle based on the fitness function.

[0035] Step 332: Initialize the individual optimal (pbest) and global optimal (gbest). For each particle, set its pbest position as its initial position and the corresponding fitness value as its initial fitness. Then, determine the global optimal by iterating through the initial fitness of all particles, selecting the particle with the best fitness, setting its initial position as the gbest position, and using its corresponding fitness as the gbest fitness. Both pbest and gbest need to store the corresponding position vector and fitness value. The former records the historical optimal state of each particle, and the latter records the current optimal state of the entire population, providing an initial reference standard for subsequent particle iteration updates.

[0036] Step 333: Iterative Update. Based on the PSO velocity update formula, and considering inertia weights, individual optimality, and global optimality, adjust the particle velocity. The iterative update velocity formula is:

[0037]

[0038] The particle position is updated based on the new velocity, and the iterative position update formula is as follows:

[0039]

[0040] in, is the inertial weight of the particle at the t-th iteration; c1 and c2 are acceleration factors, typically taken as 2; r1 and r2 are both random numbers between 0 and 1; V i t The velocity of the particle in the previous iteration; This represents the optimal position of the particle after the t-th iteration. This is the vector difference between the particle's current position and its "optimal position". This represents the globally optimal position of the entire particle swarm after the t-th iteration.

[0041] This is the vector difference between the particle's current position and its "global optimal position"; Let be the position vector of the particle at the t-th iteration. Then, based on the updated particle position... Repeat step 331 to update the fitness value. If the new fitness of the current particle is less than the fitness value of the particle's optimal position (pbest), then... - fit i If so, then update The fitness value (gbest) corresponding to the globally optimal position found by the entire particle swarm. - fit i ) equals the new fitness; if the pbest of all particles - fit i The smallest value is less than gbest - fit i Then update pbest to the corresponding pbest - fit i This is the minimum value.

[0042] At this point, the outer PSO optimization is complete. The optimization result (the optimal combination of hyperparameters) is then transmitted to the inner PSO. After a predetermined number of iterations by the inner PSO, the optimal connection parameters are finally output.

[0043] Step 4: Assign optimal hyperparameters and connection parameters to the BP neural network and train it to obtain the target constitutive model;

[0044] The BP neural network constructed in this invention is specifically designed for stress-strain curve prediction. It employs a three-layer structure of "1 input - 35 hidden layers - 1 output" (input is strain, output is stress), where the number of hidden layer nodes is determined according to the geometric pyramid rule. The network capacity and computational efficiency are optimized by balancing the geometric ratio of the input and output dimensions. The network uses Xavier initialization for weights to balance the signal propagation stability of each layer; the activation function is Leaky ReLU (negative slope 0.03), enhancing sensitivity to subtle changes in low-stress regions. Let the input of a neuron be a vector Q = (Q1, Q2, ..., Q...). m ) T (where m is the input dimension), then the neuron's output is:

[0045]

[0046] In the formula α is the weight vector of the neuron; Q is the input vector of the neuron; b is the bias; z is the net input of the activation function; σ(z) is the activation function.

[0047] The strain values ​​corresponding to the sample feature points of the stress-strain sample curves are normalized and used as the input values ​​of the neural network, while the stress values ​​corresponding to the sample feature points of the stress-strain sample curves are normalized and used as the output values ​​of the neural network. The constructed neural network is then trained. MinMaxScaler is used to scale the strain and stress to the [0, 1] interval to improve optimization stability. The stress-strain sample curve feature value normalization method is shown below:

[0048]

[0049] Where X represents the original data; X h For the processed data; X min and X max These are the minimum and maximum values ​​of the data.

[0050] The optimized BP neural network was trained using the sample database established in step 1, and then the stress-strain curves of GH4169 material under different temperatures and strain rates were predicted.

[0051] The improved two-layer PSO-BP framework in step 2 decomposes the particle swarm search space into two low-dimensional subspaces: in the inner search space, the dimension is determined only by the connection parameters, while the remaining hyperparameters determine the outer dimension. Its overall complexity is significantly lower than traditional methods, and PSO converges more easily.

[0052] The specific formula for the mean square error in step 3 is as follows:

[0053]

[0054] Where k is the number of samples in the low-stress region (strain ≤ 0.1); y ture [i] represents the true value of the i-th sample; y pred [i] represents the predicted value of the i-th sample.

[0055] The geometric pyramid rule in step 4 is as follows: from the input layer to the output layer of the neural network structure model, the number of neural network nodes in each layer decreases continuously, and the number of neural network nodes in each layer is reduced by 2 compared to the previous layer, forming a geometric pyramid structure.

[0056] Compared with existing technologies, this invention proposes a constitutive model construction method for GH4169 superalloy based on a two-layer PSO-BP neural network. The proposed improved PSO-BP enhances the model's prediction accuracy for the mechanical properties of GH4169 superalloy under complex physical loading conditions and multiple stress states, providing more reliable theoretical support for the engineering application of GH4169 superalloy. The method proposed in this invention has good versatility and can be widely applied to various superalloy materials and related experimental scenarios, especially demonstrating excellent predictive ability for the stress-strain curves of most superalloys under Hopkinson bar compression tests. Compared with the traditional PSO-BP method, the improved model in this invention achieves further breakthroughs in prediction accuracy: its coefficient of determination (R²) has been verified to be significantly higher than that of traditional PSO-BP methods. 2 The values ​​remained stable at around 0.95, and the mean square error (MSE) was controlled within 0.02, which quantitatively demonstrates the significant improvement in the overlap between the predicted curve and the experimental data curve. These results strongly reflect the effectiveness and advancement of this invention in characterizing the dynamic mechanical properties of high-temperature alloys. Attached Figure Description

[0057] Figure 1 This is a framework diagram of the specific optimization process of the two-layer PSO-BP neural network in an embodiment of the present invention;

[0058] Figure 2 This is a flowchart illustrating the steps involved in prediction using a two-layer PSO-BP neural network according to an embodiment of the present invention.

[0059] Figure 3 In this embodiment of the invention, a traditional PSO-BP neural network is used at a strain rate of 5000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0060] Figure 4 In this embodiment of the invention, a traditional PSO-BP neural network is used at a strain rate of 6000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0061] Figure 5 In this embodiment of the invention, a traditional PSO-BP neural network is used at a strain rate of 7000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0062] Figure 6 In this embodiment of the invention, a traditional PSO-BP neural network is used at a strain rate of 8000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0063] Figure 7In this embodiment of the invention, a two-layer PSO-BP neural network is used at a strain rate of 5000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0064] Figure 8 In this embodiment of the invention, a two-layer PSO-BP neural network is used at a strain rate of 6000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0065] Figure 9 In this embodiment of the invention, a two-layer PSO-BP neural network is used at a strain rate of 7000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions;

[0066] Figure 10 In this embodiment of the invention, a two-layer PSO-BP neural network is used at a strain rate of 8000 s. -1 Predicted stress-strain curves of GH4169 material at different high temperatures under different operating conditions. Detailed Implementation

[0067] To make the objectives, features, and advantages of this invention more apparent and understandable, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments. Obviously, this invention is not limited to the following embodiments, and specific implementation methods can be determined according to the technical solutions of this invention and actual circumstances. To avoid obscuring the essence of this invention, well-known methods, processes, and procedures are not described in detail.

[0068] This invention uses a backpropagation (BP) neural network as its basic framework and introduces a particle swarm optimization (PSO) algorithm to optimize and improve its prediction error, overcoming the problem of BP neural networks easily getting trapped in local optima and reducing the prediction error. This optimization scheme improves the model's prediction accuracy for the mechanical properties of GH4169 superalloy under complex physical loading conditions and multiple stress states, providing more reliable theoretical support for the engineering application of this material. The specific optimization process is as follows: Figure 1 As shown. The method of the present invention includes: obtaining stress-strain data through Hopkinson bar experiments and constructing a sample library; executing a two-layer PSO algorithm to optimize the hyperparameters and intrinsic parameters of a BP neural network; training the BP neural network using the optimized parameters to generate a pre-built model; predicting the stress-strain behavior under new working conditions, and calculating RMSE and R0. 2 To evaluate the accuracy of the model.

[0069] This invention provides a method for constructing a constitutive model of GH4169 high-temperature alloy based on a two-layer PSO-BP neural network. To demonstrate the applicability of this invention, it is applied to an example, specifically including the following steps:

[0070] S1: The stress-strain curves of GH4169 high-temperature alloy were obtained through the Hopkinson bar test, and a BP neural network training sample database was established.

[0071] Hopkinson bar experiments were conducted under preset loading pressure, different high temperatures, and different high strain rates. The experimental variables selected in this invention were: strain rate of 5000 s⁻¹. -1 6000s -1 7000s -1 8000s -1 Temperatures of 500℃, 600℃, 700℃, and 800℃ were selected. Stress-strain curves of GH4169 high-temperature alloy were obtained, and a sample database was established for training the neural network.

[0072] S2: Construct a two-layer particle swarm optimization framework (PSO) and improve the selection method of inertia weight parameters in the inner and outer layers;

[0073] S21: Two-layer PSO-BP optimization framework

[0074] A two-layer PSO-BP optimization framework is constructed, which consists of an outer PSO layer and an inner PSO layer, forming a nested loop structure.

[0075] In the outer PSO layer, each particle corresponds to a set of hyperparameter combinations, and the fitness of each particle is determined by the optimization results of the embedded inner PSO layer. The outer PSO layer updates the particle positions iteratively (adjusting the hyperparameters) and eventually searches for the hyperparameter combination that optimizes the prediction accuracy of the neural network.

[0076] The inner PSO is driven by the outer PSO. Each time the outer PSO generates a set of hyperparameters, the inner PSO is initiated for optimization. The inner particles represent the combination of connection parameters (weights and thresholds). The prediction error of the training set is used as the fitness. The optimal connection parameters under the current hyperparameter structure are obtained through iterative optimization. The optimization error is fed back to the outer layer as the basis for hyperparameter evaluation.

[0077] S22: Improved method for selecting inertia weight

[0078] Inertia weight (ω) is the core parameter of Particle Swarm Optimization (PSO). It balances global exploration and local development capabilities by adjusting the strength of particle inheritance of historical velocity: the larger the value, the stronger the particle's global search capability, which is conducive to escaping local optima; the smaller the value, the more easily the particle is guided by the group's experience, which is conducive to local fine optimization.

[0079] This invention employs an adaptive approach to the inertia weight in the particle swarm optimization algorithm. The inertia weight and the number of iterations are related to the fitness of each particle (characterized by mean squared error, MSE). The dynamic adjustment logic of the inertia weight is as follows: a larger value is used in the early stages of iteration to ensure exploration breadth, and a smaller value is used in the later stages of iteration to enhance convergence accuracy. Therefore, both the inner and outer layers ultimately obtain the minimum value obtained during the iteration process. In this invention, the inertia weight change rule is as follows:

[0080] Where, ω min and ω max The minimum and maximum inertia weight coefficients are given in advance, and are generally taken as 0.4 and 0.9, respectively. Let be the average fitness of all particles at the d-th iteration, defined as:

[0081] Let n be the fitness of a particle in the d-th iteration; n is the number of samples. Let be the minimum fitness of all particles at the d-th iteration, defined as:

[0082]

[0083] During iterative optimization, some particles will find better positions, while others will be in medium or poor positions. In the next iteration, particles in better positions are more likely to find better solutions, while the performance of particles in poorer positions further weakens. As iteration continues, the better particles will gradually approach or even reach the global optimum. Based on this, in each iteration update, the inertia weights are dynamically adjusted according to the fitness values ​​of the particles in the previous iteration.

[0084] S3: Optimize the BP neural network using the two-layer PSO algorithm to obtain the optimal hyperparameters and connection parameters (weights, thresholds);

[0085] S31: Initialize the PSO parameters for both the inner and outer layers.

[0086] In the two-layer PSO-BP framework, the parameters (number of iterations, population size, learning factor, etc.) of the inner PSO and the outer PSO are independent and need to be designed separately according to the optimization objectives and task characteristics of the two layers of PSO.

[0087] First, the outer PSO determines the search range of hyperparameters (hidden layer neurons, learning rate, and number of backpropagation training iterations) and generates specific hyperparameter values ​​(i.e., initial positions) for each particle.

[0088] To adapt to the strong nonlinearity of GH4169 and avoid insufficient or redundant neurons, the neuron search range is determined to be [10, 20]. Under high strain rates, stress data changes drastically, requiring a smaller learning rate η to avoid training oscillations. If η < 0.005, convergence is too slow; if η > 0.02, the optimal weight region will be skipped. Therefore, this invention limits the learning rate η to [0.005, 0.02]. The signal-to-noise ratio of high strain rate data is lower than that of quasi-static experiments, requiring sufficient iterations for stable model convergence. However, when the number of iterations exceeds 800, the predicted MSE of GH4169 no longer decreases but instead increases due to overfitting noise. To ensure model convergence, the number of iterations is limited to [300, 800]. To avoid local optima under complex problems and cover potentially high-quality hyperparameter combinations, the outer PSO population size is set to 20, and all particles are assigned a set of hyperparameter combinations within the aforementioned hyperparameter range.

[0089] Secondly, the inner PSO generates connection parameters for each outer particle's hyperparameters, with the connection parameter range set to the conventional [-0.5, 0.5]. The training set MSE is calculated through forward propagation of the BP network and used as the initial fitness of the inner particles, thereby updating the individual optimality and global optimality of the inner particles.

[0090] S32: Calculate the fitness of each particle in the inner and outer layers and dynamically adjust the inertia weights.

[0091] In BP neural network optimization, particle positions represent the network hyperparameters optimized in the outer layer and the connection parameters optimized in the inner layer PSO, respectively. The fitness function is used to quantify the network's prediction accuracy. This invention uses the reciprocal of the mean squared error (MSE) as the fitness metric: the inner layer PSO obtains the network's MSE on the training set by optimizing the connection parameters under the corresponding hyperparameters; the outer layer PSO quantifies the particle fitness based on the network's predicted MSE corresponding to the current hyperparameter combination. The specific calculation formulas are as follows:

[0092]

[0093] Among them, Fitness out (x) represents the fitness of the outer particles; MSE(x) represents the mean square error of the outer particles after optimization; Fitness in (ω) represents the fitness of the inner particles; MSE(ω) represents the mean square error of the inner particles after optimization.

[0094] During each iteration, the inertia weights are dynamically adjusted in the next iteration based on the fitness values ​​of the particles from the previous iteration. The core of the two-layer framework is the nested logic of "outer layer optimizing hyperparameters → inner layer optimizing connection parameters," but the iteration processes of the two PSO layers are completely independent, each possessing its own algorithm parameters. Therefore, when optimizing the neural network, the inertia weights of the inner and outer PSO layers are dynamically adjusted separately.

[0095] S33: By updating the velocity and position of the particle swarm, the optimal fitness of the swarm is output after continuous iteration. The specific steps are as follows:

[0096] S331: Calculate the initial fitness. For each particle, decode its position into the weights and biases of the BP neural network. Initialize the BP neural network and calculate the initial fitness value (MSE) for each particle based on the fitness function.

[0097] S332: Initialize the individual optimal (pbest) and global optimal (gbest). For each particle, set its pbest position as its initial position and the corresponding fitness value as its initial fitness. Then, determine the global optimal by iterating through the initial fitness of all particles, selecting the particle with the best fitness, setting its initial position as the gbest position, and using its corresponding fitness as the gbest fitness. Both pbest and gbest need to store the corresponding position vector and fitness value. The former records the historical optimal state of each particle, and the latter records the current optimal state of the entire population, providing an initial reference standard for subsequent particle iteration updates.

[0098] S333: Iterative Update. Based on the PSO velocity update formula, and considering inertia weights, individual optimality, and global optimality, the particle velocity is adjusted. The iterative update velocity formula is:

[0099]

[0100] The particle position is updated based on the new velocity, and the iterative position update formula is as follows:

[0101]

[0102] in, is the inertial weight of the particle at the t-th iteration; c1 and c2 are acceleration factors, typically taken as 2; r1 and r2 are both random numbers between 0 and 1; V i t The velocity of the particle in the previous iteration; This represents the optimal position of the particle after the t-th iteration. This is the vector difference between the particle's current position and its "optimal position". This represents the globally optimal position of the entire particle swarm after the t-th iteration. This is the vector difference between the particle's current position and its "global optimal position"; Let be the position vector of the particle at the t-th iteration. Then, based on the updated particle position... Repeat step 331 to update the fitness value. If the new fitness of the current particle is less than the fitness value of the particle's optimal position (pbest), then... - fit iIf so, then update The fitness value (gbest) corresponding to the globally optimal position found by the entire particle swarm. - fit i The new fitness is equal to the pbest-fit of all particles; i The smallest value is less than gbest - fit i Then update pbest to the corresponding pbest - fit i This is the minimum value.

[0103] At this point, the outer PSO optimization is complete. The optimization result (the optimal combination of hyperparameters) is then transmitted to the inner PSO. After a predetermined number of iterations by the inner PSO, the optimal connection parameters are finally output.

[0104] S4: Assign optimal parameters to the BP neural network and train it to obtain the target constitutive model;

[0105] The BP neural network constructed in this invention is specifically designed for stress-strain curve prediction. It employs a three-layer structure of "1 input - 35 hidden layers - 1 output" (input is strain, output is stress), where the number of hidden layer nodes is determined according to the geometric pyramid rule. The network capacity and computational efficiency are optimized by balancing the geometric ratio of the input and output dimensions. The network uses Xavier initialization for weights to balance the signal propagation stability of each layer; the activation function is LeakyReLU (negative slope 0.03), enhancing sensitivity to subtle changes in low-stress regions. Let the input of a neuron be a vector Q = (Q1, Q2, ..., Q...). m ) T (where m is the input dimension), then the neuron's output is:

[0106]

[0107] In the formula α is the weight vector of the neuron; Q is the input vector of the neuron; b is the bias; z is the net input of the activation function; σ(z) is the activation function.

[0108] The strain values ​​corresponding to the sample feature points of the stress-strain sample curves are normalized and used as the input values ​​of the neural network, while the stress values ​​corresponding to the sample feature points of the stress-strain sample curves are normalized and used as the output values ​​of the neural network. The constructed neural network is then trained. MinMaxScaler is used to scale the strain and stress to the [0, 1] interval to improve optimization stability. The stress-strain sample curve feature value normalization method is shown below:

[0109]

[0110] Where X represents the original data; X h For the processed data; X min and X max These are the minimum and maximum values ​​of the data.

[0111] The optimized BP neural network is trained using the sample database established in step S1, and then the stress-strain curves of GH4169 material under different temperatures and strain rates are predicted.

[0112] The training process of a BP neural network is as follows: First, the BP network is initialized, including setting the network structure parameters and initial weights and thresholds. Second, training samples (i.e., incident wave signals) are input into the input layer. Then, the hidden layer architecture is defined, and the number of hidden layer nodes is determined, thereby completing the initial configuration of network weights and thresholds. Based on this, the signal is transmitted to the output layer through the forward propagation mechanism to obtain predicted outputs such as reflected waves and transmitted waves. Next, the error backpropagation algorithm is used to calculate the error between the predicted output and the expected output. If the error does not meet the preset accuracy requirements, the error information is propagated back along the network to iteratively adjust the weights and thresholds of each hidden layer. Afterward, the weights and thresholds are reconfigured based on the adjusted parameters, and the process of forward propagation calculation and error backpropagation adjustment is repeated. When the error converges to the preset expected range, the training process terminates. The entire training process relies on the iterative mechanism of "forward propagation to calculate error and backpropagation to optimize parameters," enabling the network output to gradually approach the expected output, ultimately achieving effective network training. The specific steps of prediction are as follows... Figure 2 As shown.

[0113] The final prediction results are as follows Figure 3 , Figure 4 , Figure 5 and Figure 6 As shown, the predicted curves for four strain rates and four different temperature conditions are presented. The following section selects... Figure 3 (Strain rate: 5000s) -1 (Temperatures: 500℃, 600℃, 700℃, 800℃) will be explained.

[0114] This invention uses the root mean square error (RMSE) to calculate the deviation between the predicted curve and the experimental curve; simultaneously, to more objectively observe the degree of fit between the predicted and experimental curves, a dimensionless evaluation index is adopted, introducing the coefficient of determination (R²). 2 The formulas for the two indicators are shown below:

[0115]

[0116] In the formula: RMSE represents the root mean square error; This indicates the stress value at a characteristic point of the predicted curve; s represents the stress value at the characteristic point of the test curve; s represents the number of characteristic points of the stress-strain curve selected.

[0117]

[0118] In the formula: R 2 Indicates the coefficient of determination; This represents the average stress at the characteristic point.

[0119] The root mean square error (RMSE) reflects the deviation between the predicted and experimental values. A smaller RMSE value indicates a smaller deviation between the simulated and experimental values, and a better simulation result. The coefficient of determination (R²) 2 The coefficient of determination (R²) is a statistical indicator used to reflect the reliability of a regression model in explaining changes in the dependent variable. A higher R² indicates a stronger correlation between the two variables. The R² ranges from 0 to 1. 2 A value ≥0.5 indicates a strong mathematical correlation between the two. When the traditional PSO-BP neural network model predicts the stress-strain curve of GH4169 under complex working conditions, it is prone to getting trapped in local optima and insufficient feature capture, as shown in the prediction results. Figure 3 , Figure 4 , Figure 5 and Figure 6 As shown. This invention specifically addresses the above-mentioned problems by optimizing the model structure and algorithm logic, effectively improving the model's nonlinear fitting ability and prediction stability. The final prediction results of the improved model are shown below. Figure 7 , Figure 8 , Figure 9 and Figure 10 As shown; to quantify the improvement effect, Tables 1 and 2 respectively compare and analyze the accuracy of the two predictions.

[0120] Table 1. RMSE and R of the two-layer PSO-BP model 2 value

[0121]

[0122] Table 2 RMSE and R of the traditional PSO-BP model 2 value

[0123]

[0124] In Table 1, the overall RMSE is less than 0.02, R 2 The value fluctuated around 0.95; the RMSE values ​​in Table 2 were generally between [0.02, 0.12], R 2Most values ​​are below 0.8. Based on this, the constitutive model of GH4169 high-temperature alloy constructed in this embodiment using a two-layer PSO-BP neural network has been verified by results to show that, compared with the traditional PSO-BP neural network prediction method, it has achieved a significant improvement in the accuracy of characterizing the mechanical behavior and predicting the performance parameters of GH4169 high-temperature alloy.

[0125] (1) Based on the Hopkinson bar experiment, this embodiment of the invention establishes a neural network prediction model as shown in step S4. This model is applicable to most high-temperature alloy materials, and its prediction effect is not limited to predicting the stress-strain curve of the material.

[0126] (2) In the parameter optimization stage, the embodiment of the present invention adopts a two-layer PSO-BP structure that optimizes hyperparameters and "predictable parameters" (thresholds and weights) separately, which transforms the high-dimensional search space of the traditional PSO-BP into two low-dimensional search spaces, thereby increasing the prediction accuracy.

[0127] (3) Compared with traditional single prediction methods, the method described in this embodiment of the invention has higher accuracy in predicting the stress-strain curve of GH4169 material under high temperature and high strain rate conditions; the root mean square error (RMSE) between the prediction results and experimental results in the embodiment is as low as 0.0024, and the coefficient of determination (R²) is also lower. 2 With a value as high as 0.9975, it can effectively and accurately predict the stress-strain curve of GH4169 material.

[0128] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for constructing a GH4169 high-temperature alloy constitutive model based on a double-layer PSO-BP neural network, characterized in that, The method comprises the following steps: Step 1: Obtain the stress-strain curves of GH4169 high-temperature alloy under different temperatures and strain rates through Hopkinson pressure bar experiments, and establish a BP neural network training sample database; Step 2: Construct a double-layer particle swarm optimization (PSO) optimization framework, including an outer-layer PSO and an inner-layer PSO, the outer-layer PSO is used for optimizing hyperparameters, and the inner-layer PSO is used for optimizing connection parameters, and an adaptive inertia weight adjustment strategy is adopted; the connection parameters include weights and thresholds; Step 3: Optimize the hyperparameters and connection parameters of the BP neural network respectively by using the double-layer PSO, and obtain an optimal parameter combination; Step 4: Assign the optimal hyperparameters and connection parameters to the BP neural network and perform training, and finally obtain a constitutive model of the GH4169 high-temperature alloy, which is used for predicting stress-strain curves under different temperatures and strain rates.

2. The method for constructing a GH4169 high-temperature alloy constitutive model based on a double-layer PSO-BP neural network according to claim 1, characterized in that, The double-layer PSO framework in step 2 is a nested structure: The particles of the outer-layer PSO represent hyperparameter combinations, and the fitness is determined by the optimization result of the inner-layer PSO; The particles of the inner-layer PSO represent connection parameter combinations, and the fitness is measured by the mean square error (MSE) of the training set; The outer-layer PSO and the inner-layer PSO independently perform iterations, and each adopts an adaptive inertia weight adjustment strategy.

3. The method for constructing a GH4169 high-temperature alloy constitutive model based on a double-layer PSO-BP neural network according to claim 1 or 2, characterized in that, In step 2, the adaptive inertia weight adjustment strategy has the following expression: Where, ω min and ω max These are the preset minimum inertia weight coefficient and maximum inertia weight coefficient, respectively; Let be the average fitness of all particles at the d-th iteration; For a certain particle in the d-th iteration fitness; Let be the minimum fitness of all particles at the d-th iteration. The definition is: The definition is:

4. The method according to claim 1, wherein the method is characterized in that, Step 3 specifically includes: Initialize the algorithm parameters of the outer-layer PSO and the inner-layer PSO respectively, including population size, iteration number, learning factor and inertia weight range; The search range of the hyperparameters in the outer-layer PSO is set as: the number of hidden layer neurons [10, 20], the learning rate [0.005, 0.02], and the training times [300, 800]; The initial range of the connection parameters in the inner-layer PSO is set as [-0.5, 0.5]; Update the particle velocity and position through iteration, calculate and update the fitness, individual optimum and global optimum of the inner and outer layer particles respectively, and finally output the optimal hyperparameter combination and the corresponding optimal connection parameters.

5. The method of claim 1, wherein the method is characterized by: The structure of the BP neural network in step 4 is: Input layer: 1 node; Hidden layer: 35 hidden nodes, determined according to the geometric pyramid rule; Output layer: 1 node; The input is the normalized strain value, and the output is the normalized stress value; The hidden layer activation function adopts Leaky ReLU, and the negative slope is set to 0.03; The weight initialization adopts the Xavier method; Data normalization: use MinMaxScaler to scale the strain and stress to the [0, 1] interval.

6. The method of claim 1, wherein the method is characterized by: The training process in step 4 includes: Forward propagation to calculate the output; Calculate the prediction error; Error back propagation to adjust the weights and thresholds; Iterate until the error converges to a preset range.

7. The method according to claim 5, wherein the method is characterized by, The normalization processing adopts the MinMaxScale method to scale the strain and stress data to the [0, 1] interval, and the specific formula is: where X is the original data; X h is the processed data; X min and X max are the minimum and maximum values of the data.

8. The method according to claim 1, wherein the method is characterized in that, The method further comprises a model accuracy evaluation step: Root Mean Square Error, RMSE, and coefficient of determination, R 2 Quantify the degree of fit of the predicted curve to the experimental curve. The calculation formulas of the RMSE and R 2 are respectively: wherein: RMSE represents the root mean square error; represents the stress value of the characteristic point of the prediction curve; represents the stress value of the characteristic point of the test curve; s represents the number of selected stress-strain curve characteristic points; wherein: R 2 represents the coefficient of determination; represents the average stress of the feature points.

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