Optimization problem efficient solving method and device based on coupled physical neural network

Abstract

The invention discloses an optimization problem efficient solving method and device based on a coupled physical neural network, and belongs to the field of deep learning simulation solving. The method comprises the following steps: S1, inputting an optimization problem; s2, constructing accurate solution data; s3, establishing a constraint condition network and training; s4, establishing a target function network; s5, taking the objective function as a loss function, and establishing a dual-network coupling architecture; s6, training a dual-network coupling architecture; s7, outputting a predicted optimal solution result; and S8, performing deviation correction on the predicted optimal solution result by using a deviation correction method to obtain a high-precision optimal solution result. According to the method, a multi-task objective function can be quickly coped with, so that a large number of repeated solving data sets are avoided, and the target required by a product is responded more timely and accurately; meanwhile, the innovative training method not only ensures the global search capability, but also effectively balances the calculation cost and the solving precision, and provides efficient technical support for solving a complex optimization problem.

Description

Technical Field

[0001] The present invention relates to a method and device for efficiently solving optimization problems based on a coupled physics neural network, belonging to the field of deep learning simulation and solution, and is particularly suitable for the efficient solution of optimization problems based on a coupled physics neural network. Background Art

[0002] As a core technology in modern product R & D, simulation modeling plays an irreplaceable role in accurately simulating key characteristics such as product kinematics and mechanics. With the rapid development of intelligent manufacturing, this technology has been deeply integrated into the whole process of new product design and manufacturing, and has become a key support for improving R & D efficiency and product quality. It is worth noting that with the development of market economic demands, product demands show diversified characteristics. The design and optimization of a single product need to respond to multiple task objectives in a timely manner, which makes the multi-task optimization problem based on simulation modeling an urgent frontier field to be broken through.

[0003] Existing global optimization methods have significant limitations in dealing with complex engineering problems: deterministic methods are only applicable to optimization problems with obvious characteristics such as convexity or monotonicity; random methods such as degradation algorithms and genetic algorithms are limited to solving discrete points and cannot capture the complete information of continuous intervals.

[0004] Although the neural network method provides an effective way for the global search of non-linear optimization problems. The physics-informed neural network (PINN) proposed by Raissi et al. significantly reduces the dependence on sample data by embedding the control equation into the loss function. In view of the dual advantages of the physics-informed neural network in accurately fitting sample points and smoothly constraining the solution space, the present invention chooses to deeply explore along this technical route to solve the efficient and high-precision solution of non-linear optimization problems in multi-task scenarios. However, when dealing with complex optimization problems such as multi-task objectives and non-linear systems, it still faces three major challenges: (1) Computational efficiency bottleneck: Due to the specificity of task objectives, existing methods need to train each task separately, resulting in a large consumption of computing resources. (2) Global guarantee problem: How to ensure the optimal solution of the optimization problem at the global level. (3) Model training efficiency problem: The training of the neural network model relies on a large amount of data. To accurately express the optimization problem, data needs to be selected strategically.

[0005] In summary, in the design and development of products, it is often necessary to optimize the corresponding simulation model of the same existing product according to different requirements - objective functions, that is, under changing objectives, how to quickly solve the optimization problem with the same constraint conditions. Especially as the scale and parameters of the system gradually increase, the introduction of a large number of parameters will lead to extremely low solution efficiency. Summary of the Invention

[0006] In view of this, in order to solve the non-linear optimization problem in an efficient and high-precision multi-task scenario, the present invention proposes an efficient method and device for solving optimization problems based on a coupled physical neural network, aiming to propose a dual-network coupling architecture: through the collaborative work of two physical information neural networks, the constraint conditions and objective functions of the optimization problem are equivalently modeled and solved respectively, so as to be able to quickly handle multi-task objective functions and avoid a large number of repeated solution data sets. At the same time, in order to further improve the training efficiency and accuracy, a physical information neural network training method based on a genetic algorithm for the dual-network coupling architecture is proposed, which not only ensures the global search ability, but also effectively balances the computational cost and solution accuracy, providing efficient technical support for the solution of complex optimization problems.

[0007] To achieve the above object, the present invention provides the following technical solutions:

[0008] An efficient method for solving an optimization problem based on a coupled physical neural network, combined with Figure 1 , includes the following steps:

[0009] S1: A user inputs an optimization problem to be solved;

[0010] S2: Construction of an accurate solution data set for constraint conditions

[0011] S3: Using a physical information neural network to establish a constraint condition network and training it with the accurate solution data set;

[0012] S4: Using a deep learning network to establish an objective function network;

[0013] S5: Taking the objective function as a loss function and embedding the constraint condition network into the objective function network to establish a dual-network coupling architecture;

[0014] S6: Training the dual-network coupling architecture with a random training set;

[0015] S7: Using the trained objective function network to output a predicted optimal solution result;

[0016] S8: Using a bias correction method to correct the predicted optimal solution result to obtain a high-precision optimal solution result.

[0017] Further, the optimization problem is in the form of:

[0018] ;

[0019] where the independent variable , the parameter , the dependent variable on, and the function mapping , , ; corresponds to the objective function equation, corresponds to the constraint condition equation; is the parameter dimension and is a positive integer; is the dimension of the dependent variable and is a positive integer; , are the initial and termination times, , are respectively the lower bound and the upper bound of the value of the parameter .

[0020] Furthermore, the constraint condition network is composed of a deep learning network in series with the constraint condition equation; the objective function network is composed of a deep learning network in series with a parameter normalization equation; the dual-network coupling architecture is composed of the objective function network in series with the constraint condition network and then in series with the objective function equation.

[0021] Furthermore, in combination with Figure 2 , the training process of the dual-network coupling architecture described in step S6 is as follows:

[0022] S601: Randomly generate a set of sample points with dimensions ;

[0023] S602: Input the set of sample points into the deep learning network of the objective function network to obtain the predicted value ; where are the parameters to be trained of the deep learning network of the objective function network;

[0024] S603: Use the parameter normalization equation to process the predicted value to obtain the parameter value ;

[0025] S604: Input the parameter value and the independent variable into the constraint condition network to obtain the predicted value of the dependent variable ; where are the parameters to be trained of the constraint condition network ;

[0026] S605: Randomly select a slack variable within the interval , and relax the dependent variable to obtain the dependent variable value ; where , is the residual of the constraint condition network ;

[0027] S606: Take the independent variable , parameter value and dependent variable value as inputs, and calculate the loss value of the objective function as the loss function ;

[0028] S607: Use gradient descent of the loss value to adjust the parameters of the deep learning network of the objective function network

[0029] S608: Repeat steps S602 to S607 until the loss value meets the threshold to complete the training

[0030] Preferably, the method for training the constraint condition network described in step S3 is a physics-informed neural network training method based on a genetic algorithm. The physics-informed neural network training method based on a genetic algorithm is specifically as follows:

[0031] S301: Initialize the population: Randomly generate an initial population containing M individuals from the parameter interval ; ;

[0032] S302: Traverse and select parameters from , replace the parameters in with those in , and use a numerical solution method to solve the corresponding dependent variable value of the independent variable at time ;

[0033] S303: Store into the exact solution training dataset

[0034] S304: Use the exact solution training dataset to train the PINN network ;

[0035] S305: If , then generate new parameter individuals using selection operation, crossover operation and mutation operation, and repeat steps S302 to S305 until and output the trained network or reach the maximum number of iterations and report an error; where is the absolute error threshold

[0036] Furthermore, the selection operation is to take the corresponding in All individuals are directly selected as the new parameter individuals of the next generation, and the dependent variable values of these parameter individuals do not need to be numerically solved again. ; After the selection operation is completed, when the number of individuals does not reach M, the crossover operation and mutation operation are performed according to a certain proportion; the crossover operation is to pair the individuals selected in the selection operation and perform the crossover operation with a certain crossover probability to generate new individuals; the mutation operation is to generate new individuals in the parameter interval with a certain mutation probability.

[0037] Specifically, this strategy realizes the automatic identification of the high-sampling region of the residual and the weight optimization of key parameters through selection, crossover and mutation operations. While ensuring the prediction accuracy of the constraint condition network, it effectively avoids the disorderly expansion of the differential-algebraic equation data set and improves the training efficiency of stochastic optimization.

[0038] Preferably, for the convenience of calculation, all elements in can take which is greater than in a high-probability sense, which can ensure the equivalence of the solutions.

[0039] Preferably, in steps S1 to S4, the constraint condition network can be built and trained offline first, and the objective function network can be built; only the operations in steps S5 to S8 need to be adjusted according to the objective function online; this can greatly improve the calculation efficiency.

[0040] Preferably, the correction method is the Newton iteration method or the random walk method.

[0041] The present invention also discloses an electronic device, including at least one processor; and a memory communicatively connected to the at least one processor; wherein,

[0042] the memory stores a computer program executed by the at least one processor,

[0043] the computer program is executed by the at least one processor, so that the at least one processor can execute the above-mentioned method for efficiently solving optimization problems based on a coupled physical neural network.

[0044] Finally, the present invention also discloses a computer-readable storage medium, which stores computer instructions for causing a processor to execute the above-mentioned method for efficiently solving optimization problems based on a coupled physical neural network when executed.

[0045] Equivalence between the solution of the double-coupled network architecture of the present invention and the solution of the optimization problem: When the approximation accuracy of the PINN network is high enough, the corresponding residual Approaching the zero vector, then and . Thus, by introducing the slack variable afterwards, the solution formula is equivalent to solving the optimization problem.

[0046]

[0047] The beneficial effects of the present invention are as follows: The present invention provides an efficient method and device for solving optimization problems based on a coupled physical neural network, and innovatively proposes a dual-network coupling architecture: through the collaborative work of two physical information neural networks, the constraint conditions and objective function of the optimization problem are equivalently modeled and solved respectively. The decoupled expression of the constraint condition network for the nonlinear system can be generalized and called with only a single training. The objective function network, by introducing a slack variable of the global error upper limit, theoretically ensures the equivalence between the network architecture and the solution of the optimization problem. The present invention can quickly handle multi-task objective functions to avoid a large number of repeated solution data sets, making the goal of responding to product requirements more timely and accurate. To further improve the training efficiency and accuracy, the present invention also proposes a training method for physical information neural networks based on the genetic algorithm, which not only ensures the global search ability but also effectively balances the computational cost and solution accuracy, providing efficient technical support for the solution of complex optimization problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] To make the objectives and technical solutions of the present invention clearer, the following drawings are provided for the description of the present invention:

[0049] Figure 1 is the flowchart of the method of the present invention;

[0050] Figure 2 is the dual-coupling network architecture diagram of the method of the present invention;

[0051] Figure 3 is the graph of the change in the training prediction error of the physical information neural network based on the genetic algorithm in Embodiment 1 of the method of the present invention;

[0052] Figure 4 is the distribution diagram of the objective function values in the traversal process of using the branch and bound method in Embodiment 1 of the method of the present invention;

[0053] Figure 5 is the distribution diagram of the predicted objective function values of the dual-coupling architecture in Embodiment 1 of the method of the present invention;

[0054] Figure 6 is the structural schematic diagram of the electronic device in Embodiment 3 of the method of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] Example 1: The following two-component signal transduction network is a simplified model for potassium ion uptake regulation in Escherichia coli. At the same time, its mechanism of action is essentially similar to various stimulus-response mechanisms observed in bacteria, archaea, and plants. It includes the autophosphorylation of KdpD, the phosphorylation transfer reaction , the dephosphorylation of , and the formation of a transcription complex by DNA binding . The specific chemical reaction equations are described as follows:

[0056]

[0057] Among them, the superscript represents a phosphorylated species.

[0058] By modeling and extracting the mathematical model of the chemical reactions according to the concentration relationship, the constraint condition can be obtained as a differential-algebraic equation .

[0059]

[0060] Among them, the concentrations of the compounds in each reaction equation correspond to the variables and , with the unit of . The measured initial concentration of is , with the unit of . The parameter vector composed of -dimensional reaction rate constants, with the unit of . Among them, is the forward reaction rate, and is the reverse reaction rate. is the equilibrium constant of DNA binding, with the unit of

[0061] In the experiment, some parameters of the above system were measured, and , as well as 8 sets of experimental measurement data of x shown in Table 1, were obtained.

[0062] Table 1 Experimental measurement data in Example 1

[0063] l <![CDATA[t l > <![CDATA[x 1 0 > <![CDATA[x 2 0 > l <![CDATA[t l > <![CDATA[x 1 0 > <![CDATA[x 2 0 > 0 0 0 0 4 0.125 0.0063 0.0148 1 1 / 60 0.0027 0.003 5 1 / 6 0.0069 0.019 2 1 / 24 0.0068 0.0075 6 0.2 0.0067 0.0205 3 1 / 12 0.0068 0.0108 7 0.25 0.0072 0.0230

[0064] Now, it is necessary to determine 2 parameters such that the equation is highly consistent with the actual measurement on (unit: hour). That is, the objective function is:

[0065] .

[0066] The present invention provides a "method for efficiently solving optimization problems based on a coupled physical neural network", which specifically includes the following steps:

[0067] Step 1: The user inputs the constraint condition equation of the optimization problem to be solved and the objective function equation ; the electronic device processes them using lexical analysis and semantic analysis techniques, and extracts that the independent variable is time , and the dependent variable is .

[0068] Step 2: The constraint conditions are solved numerically using the ode15i solver in Matlab, and these numerical solutions are used to construct an exact solution training dataset.

[0069] Specifically, since the parameters to be optimized , have a large difference in order of magnitude, normalization processing is required.

[0070] At the same time, in the selection of the exact solution training dataset during the initial iteration, , are uniformly selected in the sample space according to a ratio of 1:5 respectively.

[0071] By analyzing the exact solution training dataset, it can be found that the value ranges of the dependent variables and are roughly as shown in Table 2. It can be seen that the scale orders of magnitude of multiple variables among them are quite different. In order to better achieve the accurate expression of the constraint conditions, normalization processing also needs to be performed according to the value ranges of the variables.

[0072] Table 2 Value range of the dependent variable in Embodiment 1

[0073] <![CDATA[x 1 > <![CDATA[x 2 > <![CDATA[y 1 > <![CDATA[y 2 > <![CDATA[y 3 > <![CDATA[y 4 > Upper bound 3.24e-2 2.1e-1 8.5e-3 100 1 4 Lower bound 0 0 0 99.9 0.975 3.79

[0074] Step 3: Use a physics-informed neural network to establish a constraint condition network and train it using the exact solution dataset.

[0075] The constraint condition network is composed of a DNN network in series with the constraint condition equation.

[0076] Specifically, a physics-informed neural network for constraint conditions is established using a DNN network with 7 intermediate layers, each layer having a size of 128. The exact solution training dataset consists of 20,000 sample points, and the in-domain training dataset consists of 20,000 randomly generated sample points.

[0077] During training, the loss function corresponding to the physics-informed neural network should be targeted, ensuring both a close distance to the training dataset of the exact solution and a relatively small error for each variable. Therefore, in this example,

[0078] Set the exact solution loss function as: ; where 、 is the exact solution;

[0079] The in-domain loss function: ;

[0080] The loss function is: , where , , represents the number of elements in the set.

[0081] The method for training the constraint condition network described above is a physics-informed neural network training method based on a genetic algorithm. The specific physics-informed neural network training method based on a genetic algorithm is as follows:

[0082] S301: Initialize the population: Randomly generate an initial population containing individuals from the parameter interval ; where ;

[0083] S302: Traverse and select parameters from Replace the parameters in and use a numerical solution method to solve the exact solution value of the dependent variable corresponding to the independent variable at time ;

[0084] S303: Store in the exact solution training dataset;

[0085] S304: Use the exact solution training dataset to train the PINN network ;

[0086] S305: If , then generate new parameter individuals using selection, crossover, and mutation operations, and repeat steps S302 to S305 until and output the trained network or reach the maximum number of iterations and report an error. Among them, .

[0087] Furthermore, the selection operation is to The corresponding ones in are directly selected as the new parameter individuals of the next generation, and the dependent variable values of these parameter individuals do not need to be numerically solved again ; After the selection operation is completed, when the number of individuals does not reach the crossover operation and mutation operation are performed in a 1:1 ratio; the crossover operation is to pair the individuals in the selection operation and perform the crossover operation with a certain crossover probability to generate new individuals; the mutation operation is to generate new individuals in the parameter interval with a certain mutation probability. The training results are as Figure 3 shown, indicating that the physical information neural network training method based on the genetic algorithm can effectively improve the approximation accuracy of the network

[0088] Step 4: Establish an objective function network using a deep learning network

[0089] The objective function network consists of a DNN network with one intermediate layer and a size of 128 in series with a parameter normalization equation

[0090] Step 5: Combine Figure 2 and use the objective function as the loss function, and embed the constraint condition network into the objective function network to establish a dual-network coupling architecture. Among them, the dotted line is the test process, the solid line is the training process, and N is the number of samples

[0091] Specifically, the loss function is .

[0092] Step 6: Train the dual-network coupling architecture using a random training set. The specific process is as follows

[0093] S601: Randomly generate a set of 500 two-dimensional sample points ;

[0094] S602: Input the sample point set into the deep learning network of the objective function network to obtain the predicted value ; Among them, is the parameter to be trained of the deep learning network of the objective function network ;

[0095] S603: Use the parameter normalization equation to process the predicted value to obtain the parameter value ;

[0096] S604: Input the parameter value and the independent variable into the constraint condition network to obtain the predicted value of the dependent variable ; Among them, The parameters to be trained for the constraint condition network ;

[0097] S605: Randomly select slack variables within the interval to relax the dependent variable to obtain the dependent variable value ; where and is the residual of the constraint condition network ; ;

[0098] S606: Use the independent variable , the parameter value and the dependent variable value as inputs to calculate the loss value of the objective function as the loss function ;

[0099] S607: Use gradient descent of the loss value to adjust the parameters of the deep learning network of the objective function network;

[0100] S608: Repeat steps S602 to S607, and complete the training after 71 iterations.

[0101] Step Seven: Use the trained objective function network to output the predicted optimal solution result.

[0102] Use randomly generated groups of data as inputs to predict and output 1 group of prediction results. On a desktop computer with an Intel(R) Core(TM) i7-14700KF CPU at 3.40 GHz and 32 GB of DDR4 memory, the entire training and prediction process takes only 0.945 seconds.

[0103] To better demonstrate the beneficial effects of the method of the present invention, a comparative experiment will be conducted using the existing technology deterministic method in this embodiment. The branch and bound -global method proposed in reference [1] finally calculates a global optimal solution through an iterative branch search process. Similarly, the constraint condition network in the dual-coupling architecture proposed by the present invention has a high enough accuracy after being trained by the PINN training method based on the genetic algorithm. The results obtained by the present invention, as shown in Figure 5 , and the results of reference [1], as shown in Figure 4 , both converge to an elliptical region inclined at 45° centered on the optimal objective function value, and the two regions highly overlap, which verifies the accuracy of the method of the present invention.

[0104] [1] K. S, I. B. Reachability analysis and deterministic global optimization of DAE models [M / OL]. Cham: Springer International Publishing, 2015: 61-116. https: / / doi.org / 10.1007 / 978-3-319-22428-2_2.

[0105] In addition, the solution surface of the optimization problem of potassium ion uptake regulation in Escherichia coli is extremely complex, and there are multiple extreme values, making it easy to fall into local optima like the method in Reference [2]. Combining the more detailed comparative experimental results given in Table 3, it is not difficult to find that the double-coupling architecture of the 𝜖-global method and the method of the present invention can avoid falling into local optima and is an optimization method in a global sense.

[0106] [2] Kremling, A, Heermann, R, Centler, F, et al. Analysis of two-component signal transduction by mathematical modeling using the kdpd / kdpesystem of escherichia coli. [J / OL]. Bio Systems, 2004, 78(1–3): 23-37. DOI:10.1016 / j.biosystems.2004.06.003.

[0107] Table 3 Comparison of calculation results with existing optimization solution methods

[0108] Method Parameter obj 𝜖 - global method [1] (4.31348e - 03, 162.9297) 0.029096 Method of Document [2] (2.9e - 3, 90) 0.0712 Predicted solution of the method of the present invention (4.4881e - 03, 1.7231e + 02) 0.0296 Corrected bias exact solution of the present invention (4.32191e - 03, 1.632459e + 02) 0.029089

[0109] Step Eight: Use the bias correction method to correct the predicted optimal solution result to obtain a high-precision optimal solution result, and feedback it to the user to guide experimental analysis.

[0110] The random walk method is used to approximate the global optimal solution. Starting from the predicted solution, the maximum walk radius is 10% of the parameter space radius. After 300-step walk iterations, a global optimal solution better than the 𝜖-global method of branch and bound is obtained, as shown in Table 3.

[0111] For training the constraint condition network, due to the high precision requirements, an offline training mode can be adopted. For online calculation, only the objective function network needs to be trained and the line prediction solved (taking 0.945 seconds), and the predicted approximate solution needs to be corrected (taking 0.469 seconds). The entire process takes about 1 second in total. Compared with the 52 seconds taken by the branch and bound 𝜖 - global method for the entire solution process, the online calculation process of the method of the present invention is more efficient.

[0112] Embodiment 2: To illustrate the change of the calculation time of the method of the present invention with the dimension and show the beneficial effects of the method of the present invention. Combining with reference [3], consider the following example of the optimization solution problem of a first-order linear system commonly found in dynamic simulation:

[0113]

[0114] Among them, , is an n - dimensional unit diagonal matrix, is an n - dimensional parameter vector, .

[0115] When the objective function is variable, in order to efficiently solve the optimization system multiple times. The present invention provides a "method for efficiently solving optimization problems based on a coupled physical neural network", which specifically includes the following steps:

[0116] Step 1: The user inputs the constraint condition equation and the objective function equation of the optimization problem to be solved; the electronic device processes them using lexical analysis and semantic analysis techniques to extract the independent variables and dependent variables among them.

[0117] Step 2: The constraint conditions use the ode45 solver in Matlab to obtain numerical solutions, and use these numerical solutions to construct an exact solution training dataset.

[0118] Step 3: Use a physics - informed neural network to establish a constraint condition network and train it using the exact solution dataset. Since the system is linearly simple and the network has a good expression effect, the training method of the physics - informed neural network based on the genetic algorithm is not adopted here for training. The traditional training method can be directly used.

[0119] Since is invertible and all eigenvalues are negative, the constraint condition system is stable. As the dimension of the equation increases, the dimensions of the dependent variables and parameters will also increase accordingly. To achieve high - precision expression, the training set also needs to be adjusted with the dimension. Specifically, the initial condition training dataset , the exact solution training dataset , the in - domain training dataset , the number of random sample points of the objective function network is 500. Further, the constraint condition network is set as a DNN network with 6 intermediate layers and a size of 128.

[0120] Step 4: Establish an objective function network using a deep learning network.

[0121] The objective function network is composed of a DNN network with 1 intermediate layer and a size of 8 in series with a parameter normalization equation.

[0122] Step 5: Combine Figure 3 , use the objective function as the loss function, and embed the constraint condition network into the objective function network to establish a dual-network coupling architecture.

[0123] Specifically, the loss function is .

[0124] Step 6: Train the dual-network coupling architecture using a random training set. The specific process is as follows:

[0125] The training process of the dual-network coupling architecture described in Step S6 is as follows:

[0126] S601: Randomly generate a -dimensional sample point set ; ;

[0127] S602: Input the sample point set into the deep learning network of the objective function network to obtain a predicted value ; where is the parameter to be trained of the deep learning network of the objective function network ;

[0128] S603: Use the parameter normalization equation to process the predicted value to obtain a parameter value ;

[0129] S604: Input the parameter value and the independent variable into the constraint condition network to obtain a predicted value of the dependent variable ; where is the parameter to be trained of the constraint condition network ;

[0130] S605: Randomly select a slack variable within the interval to relax the dependent variable to obtain a dependent variable value ; where, for the convenience of calculation, all elements in can take , ;

[0131] S606: Take the independent variable , parameter value and dependent variable value as inputs, and calculate the loss value when the objective function is the loss function ;

[0132] S607: Use gradient descent of the loss value to adjust the parameters of the deep learning network of the objective function network;

[0133] S608: Repeat steps S602 to S607 until the loss value meets the threshold setting to complete the training.

[0134] Step Seven: Use the trained objective function network to output the predicted optimal solution result.

[0135] Step Eight: Use the Newton iteration method to correct the predicted optimal solution result to obtain a high-precision optimal solution result.

[0136] A comparative experiment was conducted with the method in [3] on a desktop computer with an Intel(R) Core(TM) i7-14700KF CPU, 3.40 GHz, and 32 GB of DDR4 memory. The average time consumption of 10 repeated experiments (unit: seconds) was taken. The specific experimental results are shown in Table 4.

[0137] Table 4 Performance comparison experimental results of Example 2

[0138] n Training time consumption Prediction time consumption Corrected bias time consumption Branch and bound time consumption Predicted obj Predicted solution Optimal solution 2 96.1 0.6 0.003 1.4 6.50e-08 [0.5714,0.4541] [0.5721,0.4544] 3 435.2 1.2 0.003 8.1 1.17e-07 [0.6325,0.5137,0.4474] [0.6330,0.5141,0.4481] 4 1452.3 1.3 0.011 19.4 1.28e-07 [0.6751,0.5564,0.4902,0.4437] [0.6771,0.5572,0.4904,0.4444] 6 5991.9 1.5 0.005 90.6 8.07e-07 [0.7396,0.6170,0.5484,0.5014,0.4667,0.4392] [0.7404,0.6187,0.5506,0.5038,0.4685,0.4403] 8 18424.8 2.7 0.013 934.6 1.60e-07 [0.7852,0.6633,0.5949,0.5473,0.5109,0.4823,0.4578,0.4372] [0.7859,0.6636,0.5949,0.5474,0.5113,0.4825,0.4582,0.4377] 10 53472.4 4.8 0.008 6083.5 2.92e-07 [0.8241,0.6981,0.6285,0.5809,0.5443,0.5148,0.4902,0.4692,0.4513,0.4348] [0.8241,0.6984,0.6287,0.5811,0.5445,0.5150,0.4904,0.4694,0.4515,0.4350]

[0139] [3] Bajaj, Ishan, Faruque HM. Global dynamic optimization using edge-concave under estimator [J / OL]. Journal of Global Optimization, 2020, 77(3):487-512. https: / / doi.org / 10.1007 / s10898-020-00883-2.

[0140] Performance analysis: The computational time of the branch and bound method increases exponentially with the dimension of the problem. When the dimension exceeds 6, its response speed can no longer meet the real-time optimization requirements. In sharp contrast, the dual-coupled network architecture proposed in the present invention shows significant advantages: although the offline training phase requires a higher computational cost, the total time consumption of its online prediction phase (prediction time + correction time) is approximately linearly related to the dimension, and the computational efficiency is improved by at least one order of magnitude compared to the branch and bound method. In particular, in multi-objective optimization scenarios (such as products with rich personalized requirements), as the number of objective functions increases, the average total time consumption for a single solution of this architecture (training time + online time) shows a decreasing trend, and ultimately achieves cross-order-of-magnitude transcendence in overall efficiency.

[0141] Accuracy analysis: Experimental verification shows that the average deviation rate between the predicted solution of the architecture of the present invention and the theoretical optimal solution is basically lower than This distance can be further reduced by increasing the random sample points and the training accuracy of the constraint conditions. It is worth noting that the prediction solution without bias correction basically meets the actual accuracy requirements, which provides an effective accuracy-efficiency trade-off solution for scenarios with strict real-time requirements.

[0142] Embodiment 3: For the scenario of embodiment 1, Figure 6 The electronic device 10 is a schematic diagram of a structure that can be used to implement an embodiment of the present invention. The electronic device is intended to represent various forms of digital computers, such as laptops, desktop computers, workstations, personal digital assistants, servers, blade servers, mainframe computers, and other suitable computers.

[0143] Electronic devices may also represent various forms of mobile devices, such as personal digital assistants, cellular phones, smart phones, wearable devices (such as helmets, glasses, watches, etc.) and other similar computing devices. The components shown in the present invention, their connections and relationships, and their functions are only examples and are not intended to limit the implementation of the present invention described and / or required in the present invention.

[0144] like Figure 3As shown, the electronic device 10 includes at least one processor 11 and a memory communicatively connected to the at least one processor 11, such as a read-only memory (ROM) 12, a random access memory (RAM) 13, etc. Among them, the memory stores a computer program executable by the at least one processor. The processor 11 can execute various appropriate actions and processes according to the computer program stored in the read-only memory (ROM) 12 or the computer program loaded from the storage unit 18 into the random access memory (RAM) 13. In the RAM 13, various programs and data required for the operation of the electronic device 10 can also be stored. The processor 11, the ROM 12, and the RAM 13 are connected to each other through a bus 14. The input / output (I / O) interface 15 is also connected to the bus 14.

[0145] Multiple components in the electronic device 10 are connected to the I / O interface 15, including: an input unit 16, such as a keyboard, a mouse, etc.; an output unit 17, such as various types of displays, speakers, etc.; a storage unit 18, such as a disk, an optical disc, etc.; and a communication unit 19, such as a network card, a modem, a wireless communication transceiver, etc. The communication unit 19 allows the electronic device 10 to exchange information / data with other devices through a computer network such as the Internet and / or various telecommunication networks.

[0146] The processor 11 can be various general-purpose and / or special-purpose processing components with processing and computing capabilities. Some examples of the processor 11 include but are not limited to a central processing unit (CPU), a graphics processing unit (GPU), various dedicated artificial intelligence (AI) computing chips, various processors running machine learning model algorithms, a digital signal processor (DSP), and any appropriate processor, controller, microcontroller, etc. The processor 11 executes the various methods and processes described above, such as an efficient solution method for an optimization problem based on a coupled physical neural network.

[0147] In some embodiments, the efficient solution method for an optimization problem based on a coupled physical neural network can be implemented as a computer program, which is tangibly contained in a computer-readable storage medium, such as the storage unit 18. In some embodiments, part or all of the computer program can be loaded and / or installed onto the electronic device 10 via the ROM 12 and / or the communication unit 19. When the computer program is loaded into the RAM 13 and executed by the processor 11, one or more steps of the efficient solution method for an optimization problem based on a coupled physical neural network described above can be executed. Alternatively, in other embodiments, the processor 11 can be configured to execute the efficient solution method for an optimization problem based on a coupled physical neural network in any other appropriate manner (for example, by means of firmware).

[0148] The various embodiments of the systems and techniques described above in the present invention can be implemented in digital electronic circuitry, integrated circuit systems, field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), application specific standard products (ASSPs), systems on a chip (SOCs), complex programmable logic devices (CPLDs), computer hardware, firmware, software, and / or combinations thereof. These various embodiments can include: being implemented in one or more computer programs that can be executed and / or interpreted on a programmable system including at least one programmable processor, which can be a special-purpose or general-purpose programmable processor that receives data and instructions from a storage system, at least one input device, and at least one output device, and transmits the data and instructions to the storage system, the at least one input device, and the at least one output device.

[0149] The computer programs for implementing the methods of the present invention can be written in any combination of one or more programming languages. These computer programs can be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus, such that the computer programs, when executed by the processor, cause the functions / operations specified in the flowchart and / or block diagram to be implemented. The computer programs can be executed entirely on the machine, partially on the machine, as a stand-alone software package partially on the machine and partially on a remote machine, or entirely on a remote machine or server.

[0150] In the context of the present invention, a computer-readable storage medium can be a tangible medium that can contain or store a computer program for use by or in connection with an instruction execution system, apparatus, or device. The computer-readable storage medium can include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. Alternatively, the computer-readable storage medium can be a machine-readable signal medium. More specific examples of the machine-readable storage medium would include electrical connections based on one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing.

[0151] To provide interaction with a user, the systems and techniques described herein can be implemented on an electronic device having: a display device for displaying information to the user (e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor); and a keyboard and a pointing device (e.g., a mouse or a trackball) through which the user can provide input to the electronic device. Other kinds of devices can also be used to provide interaction with the user; for example, the feedback provided to the user can be any form of sensory feedback (e.g., visual feedback, auditory feedback, or tactile feedback); and the input received from the user can be in any form (including acoustic input, voice input, or tactile input).

[0152] The systems and techniques described herein can be implemented in a computing system including backend components (e.g., as a data server), or a computing system including middleware components (e.g., an application server), or a computing system including frontend components (e.g., a user computer having a graphical user interface or a web browser through which the user can interact with an implementation of the systems and techniques described herein), or a computing system including any combination of such backend components, middleware components, or frontend components. The components of the system can be interconnected to each other by digital data communication in any form or medium (e.g., a communication network). Examples of communication networks include: local area network (LAN), wide area network (WAN), blockchain network, and the Internet.

[0153] The computing system can include a client and a server. The client and the server are generally remote from each other and typically interact through a communication network. The client-server relationship is created by computer programs running on the respective computers and having a client-server relationship with each other. The server can be a cloud server, also known as a cloud computing server or a cloud host, which is a host product in the cloud computing service system and solves the defects of difficult management and weak business scalability existing in traditional physical hosts and VPS services.

[0154] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined by the claims of the present invention.

Claims

1. An efficient solution method for optimization problems based on coupled physical neural networks, characterized by: The method comprises the following steps: S1: The user inputs the optimization problem to be solved; S2: Construction of exact solution dataset for constraints S3: Use the physical information neural network to establish the constraint network and train it using the exact solution data set; S4: Use deep learning network to establish the target function network; S5: Take the objective function as the loss function, embed the constraint network into the objective function network to establish a dual network coupling architecture; S6: Training the dual network coupling architecture using random training sets; S7: Use the trained objective function network output to predict the optimal solution result; S8: Use the correction method to correct the predicted optimal solution result to obtain a high-precision optimal solution result; The optimization problem is as follows: ； Among them, the independent variable ,parameter , The dependent variable on , function mapping , , ; Corresponding to the objective function equation, The corresponding constraint equation is: is the parameter dimension, which is a positive integer; is the dimension of the dependent variable, which is a positive integer; , are the initial and final moments, , The parameters are The lower and upper bounds of the value of ; The constraint network is composed of a deep learning network connected in series with a constraint equation; the objective function network is composed of a deep learning network connected in series with a parameter normalization equation; the dual network coupling architecture is composed of an objective function network connected in series with a constraint network and then connected in series with an objective function equation.

2. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 1 is characterized in that: The method for training the constraint condition network described in step S3 is a physical information neural network training method based on a genetic algorithm. The physical information neural network training method based on a genetic algorithm is specifically as follows: S301: Initialize the population: Randomly select from the parameter interval Generates The initial population of individuals ; S302: Traverse from Select parameters in Will Parameters in Substitute and use numerical solution method to solve the independent variable The value of the dependent variable corresponding to the time ; S303: Store into the exact solution training data set; S304: Using the exact solution training data set to train the PINN network Conduct training; S305: If , then use selection operation, crossover operation and mutation operation to generate new parameter individuals , and repeat steps S302 to S305 until And output the trained network Or the maximum number of iterations is reached and an error is reported; is the absolute error threshold.

3. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 2 is characterized in that: The selection operation is to Corresponding All individuals are selected directly as the new parameter individuals of the next generation. These parameter individuals do not need to be numerically solved for the dependent variable values ​​again. ; After the selection operation is completed, the number of individuals does not reach When the crossover operation and mutation operation are performed in proportion, the crossover operation is to pair the selected individuals and then perform the crossover operation with a certain crossover probability to generate new individuals; the mutation operation is to perform the crossover operation with a certain mutation probability in the parameter interval Generate new individuals.

4. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 1 is characterized in that: The training process of the dual network coupling architecture described in step S6 is: S601: Randomly generated indivual The sample point set of ; S602: Collect sample points Input into the deep learning network of the objective function network to get the predicted value ;in, A deep learning network with a target function network The parameters to be trained; S603: Using parameter normalization equation For the predicted value Process and get the parameter value ; S604: Set the parameter value and independent variables Input into the constraint network to get the predicted value of the dependent variable ;in, Constraint network The parameters to be trained; S605: In the interval Randomly select slack variables within , relax the dependent variable to get the value of the dependent variable ;in, , Constraint network The residual of S606: The independent variable , parameter value and the dependent variable value As input, the objective function is calculated as the loss value of the loss function ; S607: Using the gradient descent of the loss value, the parameters of the deep learning network of the objective function network are adjusted. Make adjustments; S608: Repeat steps S602 to S607 until the loss value meets the threshold setting and the training is completed.

5. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 4 is characterized in that: For ease of calculation, All elements in ,in More likely than , which can ensure the equivalence of the solutions.

6. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 1 is characterized in that: The steps S1 to S4 are performed offline to first build and train the constraint network and the objective function network; when online, only the operations of steps S5 to S8 need to be adjusted according to the objective function; this can greatly improve the computing efficiency.

7. The method for efficiently solving optimization problems based on coupled physical neural networks according to claim 1 is characterized in that: The deviation correction method is Newton iteration method or random walk method.

8. An efficient optimization problem solving device based on coupled physical neural network applied to any one of claims 1 to 7, characterized in that: The device for efficiently solving optimization problems based on coupled physical neural networks is an electronic device, comprising at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program executed by the at least one processor, and the computer program is executed by the at least one processor so that the at least one processor can execute the above-mentioned method for efficiently solving optimization problems based on coupled physical neural networks.

9. A computer-readable storage medium as claimed in any one of claims 1 to 7, characterized in that: The computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to implement the above-mentioned efficient solution method for optimization problems based on coupled physical neural networks when executed.

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