A structure-aware dual-network collaborative intelligent planning method and device

Abstract

The application provides a structure-aware dual-network collaborative intelligent planning method and device, the method comprises the following steps: transforming a constraint optimization problem; decomposing the transformed constraint optimization problem into a constraint problem and a precision problem; assigning the precision problem to a precision enhancement neural network; assigning the constraint problem to a constraint satisfaction neural network; layering the precision enhancement neural network to the constraint satisfaction neural network through distillation to obtain an optimization model; and solving the constraint optimization problem by using the optimization model. The framework SPDC proposed by the application is used to solve the constraint optimization problem of a complex structure. The SPDC framework integrates three core components of problem reconstruction, knowledge extraction and collaborative solving, and improves the solving efficiency and constraint satisfaction degree through structured guidance and model interaction.

Description

Technical Field

[0001] This invention relates to the field of constraint optimization technology, specifically to a structure-aware dual-network collaborative intelligent planning method and apparatus. Background Technology

[0002] In industrial applications such as resource scheduling, power system optimization, and intelligent control, constrained optimization problems are crucial. Traditional numerical methods have limited efficiency, while neural network technology faces an inherent trade-off between constraint satisfaction and solution accuracy. Existing methods face two major challenges: first, individual network architectures suffer from task conflicts, requiring simultaneous optimization of constraint compliance and accuracy for the same parameters; second, they neglect the structural separability of constraint equations, leading to high-dimensional redundancy and increasing the difficulty of neural network processing. Existing neural network solvers often treat different types of constraints (such as linear equality and nonlinear inequality) uniformly, ignoring the structural separability of linear constraints. This prevents simplification through dimensionality reduction, forcing the network to learn redundant high-dimensional features and increasing training difficulty. While some studies have attempted to utilize convex constraint structures (such as offline projection onto a low-dimensional subspace), these rely on initial feasible solutions and cannot collaborate end-to-end with the neural network, limiting generalization capabilities.

[0003] In summary, the existing technology has the following problems: how to balance constraint satisfaction and solution accuracy in optimization, and further improve the solution efficiency and accuracy. Summary of the Invention

[0004] The purpose of this invention is to address the problem of how to provide theoretical support for attitude observability in monocular pose estimation and improve the accuracy of pose estimation.

[0005] To this end, in one aspect, embodiments of the present invention provide a structure-aware dual-network collaborative intelligent planning method, the method comprising the following steps:

[0006] Transform the constrained optimization problem;

[0007] The transformed constrained optimization problem is decomposed into a constraint problem and an accuracy problem;

[0008] Assign the accuracy problem to an accuracy-enhancing neural network;

[0009] Assign the constraint problem to a constraint satisfaction neural network;

[0010] The precision enhancement neural network is distilled into the constraint satisfaction neural network through hierarchical distillation to obtain the optimized model;

[0011] The constrained optimization problem is solved using the optimization model described above.

[0012] On the other hand, embodiments of the present invention also provide a structure-aware dual-network collaborative intelligent planning device, including...

[0013] The transformation unit is used to transform constrained optimization problems.

[0014] Decomposition units are used to decompose the transformed constrained optimization problem into a constraint problem and a precision problem;

[0015] The first allocation unit is used to assign accuracy problems to the accuracy enhancement neural network;

[0016] The second allocation unit is used to allocate the constraint problem to the constraint satisfaction neural network;

[0017] The modeling unit is used to distill the accuracy-enhancing neural network into a constraint-satisfying neural network through hierarchical distillation to obtain an optimized model.

[0018] An optimization unit is used to solve the constrained optimization problem using the optimization model.

[0019] The above technical solution has the following beneficial effects: The structure-aware module of this invention relaxes the problem by utilizing the inherent linear constraints and projects it into an affine space for decision-making, thereby transforming the original problem into a form that is easier for neural networks to solve. Comparative experimental results clearly show that after introducing structure awareness, the constraint violations of the learning method are reduced by more than 80%. In addition, the accuracy gap of the DC3 method is also reduced by 7%. This is mainly because, when dealing with linear equality constraints, unlike the "Complete" operation in the original DC3, the structure-aware method enables the neural network to output a complete solution.

[0020] The SPDC framework proposed in this invention is used to solve constrained optimization problems with complex structures. The SPDC framework integrates three core components: problem reconstruction, knowledge extraction, and collaborative solving. Through structured guidance and model interaction, it improves solution efficiency and constraint satisfaction. Attached Figure Description

[0021] Figure 1 This is a flowchart of a structure-aware dual-network collaborative intelligent planning method provided in an embodiment of the present invention;

[0022] Figure 2 This is a structural schematic diagram of a structure-aware dual-network collaborative intelligent planning device provided in an embodiment of the present invention;

[0023] Figure 3 This is a schematic diagram of the SPDC framework provided in an embodiment of the present invention;

[0024] Figure 4 This is a schematic diagram comparing the optimization method provided in the embodiments of the present invention with a comparative example; Detailed Implementation

[0025] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0026] This invention comprises a problem structure-aware module and a hierarchical distillation framework. The process begins with the structure-aware module, which processes the original problem data by introducing relaxation variables and transforming affine inequality constraints into affine equations. Subsequently, projection operations reduce the problem to the affine space, generating a new dataset characterized by lower dimensionality, fewer equations, and improved applicability of the neural network solver. Based on the reconstructed problem, a Precision Enhancement Network (PEN) is pre-trained to establish an initial mapping for high-precision solutions. Following this, a Constraint Satisfaction Network (CSN) is trained using the same dataset, and high-dimensional features from the PEN are dynamically injected through the hierarchical distillation mechanism. This achieves simultaneous optimization of constraint satisfaction and computational accuracy. In the final deployment phase, only the trained CSN needs to be invoked, ensuring that problem constraints are satisfied while maintaining high computational accuracy.

[0027] This invention addresses the problem of minimizing an objective function under constraints of equality and inequality. As decision variables, Let x be the problem parameters. Each x is a specific problem instance. For a given variable, the goal is to find the corresponding minimization objective function while satisfying all constraints. Theoretically, it can be nonlinear or nonconvex.

[0028] (1)

[0029] Affine constraints, in particular, are a common and important type of constraint, frequently found in problems such as resource allocation and path planning. These constraints exhibit good convexity and are easier to handle. Therefore, this invention further classifies problem constraints into affine constraints and non-affine constraints. The problem form is as follows:

[0030] Constraints (2a) and (2b) are affine constraints that may be included in the problem, where the matrix... GE Constraints (2c) and (2d) are non-affine constraints. , y is the independent variable. Therefore, the objective of this invention is to train a neural network that directly maps problem instances to their optimal solutions in an end-to-end manner. Neural networks need to predict a value that minimizes the objective function. A solution that strictly satisfies all constraints at the same time.

[0031] Supervised learning faces two major challenges in solving constrained optimization problems. First, obtaining high-quality supervision signals (i.e., ground truth optimal solutions) is often difficult and expensive. Second, the goal of supervised learning is to minimize the difference between predictions and provided labels, which is fundamentally different from the goal of truly learning to solve optimization problems. Therefore, supervised methods often struggle to satisfy constraints because they fail to truly capture the underlying problem structure; even small inaccuracies in prediction can propagate to cause constraint violations.

[0032] In contrast, unsupervised learning methods do not require solving instances. Instead, their goal is to learn the problem structure, including constraints and the objective function, directly from the problem statement itself. A common strategy for constrained optimization in unsupervised learning is to incorporate a penalty function into the loss function, enabling the neural network to capture both the objective and constraint structures simultaneously.

[0033] (3)

[0034] Let $\mathbf{ ... Related.

[0035] In embodiments of the present invention, such as Figure 1 A structure-aware dual-network collaborative intelligent planning method is provided, the method comprising the following steps:

[0036] S101: Transform the constrained optimization problem; structure-aware identification and utilization of the structural features in the problem, reformulating it into a more suitable representation, making it more suitable for neural network-based solvers. To illustrate the practical applicability of this method, this invention studies affine constraints, a class of mathematical relations widely used in mathematical optimization, computational geometry, and engineering systems. Specifically, for constrained optimization problems, constraints can be divided into affine equality, affine inequality, non-affine equality, and non-affine inequality constraints, and are written as formula (2).

[0037] First, introduce slack variables. Transform the affine inequality into an affine equality.

[0038] (5)

[0039] Perform straightforward variable substitution on the relaxation problem, specifically defining , the decision variables y and S and the slack variables are re-expressed as = , where . Therefore, the transformed affine equation can be combined with the original equation, expressed as: (6)

[0040] In the above expression

[0041] Then the original problem can be reformulated as the following formula.

[0042]

[0043] In equation (7), = , = , , are all functions of i, and the concavity and convexity of f are not restricted.

[0044] According to the definition of the null space, let N denote the null space of, i.e., null(null)= and 0. The feasible solution of the optimization problem must lie in the solution space of, which can be regarded as a translation of the solution space. Therefore, the solution of can be expressed as , where is a particular solution to the original problem. To determine , Rayen et al. (Tordesillas et al., 2023) adopted the pseudoinverse method. However, this method is not suitable for ill-posed problems, requires a full-rank matrix, and has poor robustness. In contrast, the QR decomposition method computes a basis for the null space, making it very suitable for ill-posed problems, rank-deficient matrices and sparse scenarios. It provides greater numerical stability and robustness, avoids direct matrix inversion, and achieves higher computational efficiency.

[0045] Therefore, the present invention uses the QR decomposition method to determine the matrix and N; specifically, QR is performed on QR decomposition can yield , where Q is an orthogonal matrix and R is an upper triangular matrix. If true, then the null space matrix is ​​considered to be 200 elements. And the following formula can be derived.

[0046] (8)

[0047] Based on the above derivation, the original problem is projected onto the affine space defined by the affine constraints, and transformed into a problem about... The optimization problem is expressed as: (9)

[0048] Z is the independent variable in the affine space;

[0049] To ensure consistency with (1) in form, (9) is restated as follows: (10)

[0050] In equation (10) g(-) = g(T(-)) ) , ) = , ) g" (-) ∈ Rg are all The function. Therefore, SP senses and utilizes the affine constraints in the original data (1) and finally transforms it into the form (10), which has fewer equality constraints, lower dimensionality, and is more suitable for neural network solutions.

[0051] S102: Decompose the transformed constrained optimization problem into a constraint problem and a precision problem;

[0052] S103: Assign the accuracy problem to the accuracy enhancement neural network;

[0053] PEN is a network cluster designed to capture finer-grained or more precise knowledge during problem-solving without imposing strict constraints on the feasibility of the solution. The network is pre-trained to acquire detailed knowledge relevant to solving constrained optimization problems. To facilitate rapid acquisition of pre-trained models and reduce pre-training costs, this invention employs a self-supervised neural network solver. As an accuracy enhancement network, the solver uses a penalty function as its loss function and employs... (A trainable parameter) controls the strength of the penalty.

[0054] (11)

[0055] It is worth noting the penalty coefficient of the loss function. and Set to a relatively small value, allowing Be more proactive and more focused on capturing The distribution characteristics of the expert network. In a given instance, the expert network The result of reasoning It may lead to a violation of constraints, but it is more likely to retain them. More related and richer information.

[0056] The constraint satisfaction network is implemented as a network ensemble. Compared to precision-enhancing networks that emphasize solution accuracy, this ensemble is designed to prioritize rigorous feasibility, ensuring that all output solutions fully satisfy the imposed constraints—regardless of numerical optimality. This invention argues that constraint satisfaction is paramount in constrained optimization problems. Any solution that violates the constraints, no matter how precise it is with respect to the objective function, is considered infeasible and therefore lacks practical value. Therefore, the task of the constraint satisfaction network is to generate the final solution, as it has the capability to guarantee feasibility. To achieve this, this invention integrates DC3, a self-supervised learning framework developed by Donti et al., which guarantees constraint satisfaction through a dual mechanism of differentiable completion and correction. These operations aim to project the intermediate outputs of the neural network back to the feasible region, thereby ensuring that the final prediction conforms to all problem constraints. Furthermore, to improve the quality of the solution without compromising feasibility, the constraint satisfaction network is trained through a dynamic ensemble of precision-related knowledge extracted from the precision-enhancing network. This strategy enables the network to learn not only efficient but also solutions that are closer to optimal feasibility.

[0057] S104: Assign the constraint problem to a constraint satisfaction neural network;

[0058] Assuming the problem is not overdetermined, for the independent variable... Training neural networks Output ITT variable entries Then according to Solving the remaining implicit equation system composed of equality constraints entries, procedures using functions This is indicated. Then, all variables are processed to satisfy the equality constraints. ,in for and . .

[0059] (12)

[0060] Since solutions derived through the equation-solving process may lack analytical differentiability, and neural networks require gradient-based optimization, the DC3 method, as a differentiable solver, mitigates this challenge by strategically implementing the implicit function theorem to maintain mathematical rigor and computational traceability (Donti et al., 2021).

[0061] DC3 introduces a gradient-based correction mechanism. via the network Adjust output This involves moving the gradient in the direction associated with the violation of the inequality constraint. The aim of this process is to guide the solution into a region where the constraint violation is less severe. Specifically...

[0062] (13)

[0063] S105: The accuracy enhancement neural network is distilled into the constraint satisfaction neural network through hierarchical distillation to obtain the optimized model;

[0064] Constraint-Saving Network It is responsible for inferring the final solution. To enhance its performance, it comes from a precision enhancement network. The precision knowledge is extracted into an auxiliary network. The auxiliary network is trained to integrate constraint-related knowledge and accuracy-related knowledge. This is achieved through an output layer distillation strategy, where... The loss function is defined as follows: (14)

[0065] set up For a given instance, the constraints satisfy the network. Output , For precision enhancement networks The corresponding output. The enhanced signal is weighted by coefficients to emphasize the importance of precision information in the distillation process.

[0066] S106: Solve the constrained optimization problem using the optimization model.

[0067] The transformation of the constrained optimization problem specifically includes:

[0068] By introducing slack variables, the affine constraint inequality is transformed into an affine constraint equality.

[0069] The original constrained optimization problem is projected onto the affine space defined by the affine constraints and transformed into an optimization problem with respect to the independent variables of the affine space.

[0070] The optimization model is as follows: ;

[0071] in, To constrain the network The output, For precision enhancement networks The corresponding output; MAE uses the mean absolute error loss as the objective function for training the neural network; MSE uses the mean squared error loss as the objective function for training the neural network; What does it represent? What does it represent? What does it represent? What does it represent? What does it represent? What do they represent? ReLU is the activation function; KL is the divergence loss.

[0072] Assigning accuracy issues to accuracy-enhancing neural networks specifically includes:

[0073] The precision-enhancing neural network is pre-trained;

[0074] Self-supervised neural network solver As a precision enhancement network;

[0075] The self-supervised neural network solver uses a penalty function as its loss function and controls the intensity of the penalty using training parameters.

[0076] The accuracy-enhancing neural network is distilled into a constraint-satisfying neural network through hierarchical distillation to obtain an optimized model; including:

[0077] The pre-generated dataset is simultaneously input into both the accuracy-enhancing neural network and the constraint-satisfying neural network;

[0078] Then, solve for the precision-enhancing neural network and the constraint-satisfying neural network;

[0079] Based on the solutions of the precision enhancement neural network and the constraint satisfaction neural network, the constraint satisfaction neural network is updated using a gating function.

[0080] This invention also provides a structure-aware dual-network collaborative intelligent planning device, such as... Figure 2 As shown, it includes:

[0081] Transformation unit 21 is used to transform constrained optimization problems;

[0082] Decomposition unit 22 is used to decompose the transformed constrained optimization problem into a constraint problem and a precision problem;

[0083] The first allocation unit 23 is used to allocate the accuracy problem to the accuracy enhancement neural network;

[0084] The second allocation unit 24 is used to allocate the constraint problem to the constraint satisfaction neural network;

[0085] Modeling unit 25 is used to extract the precision enhancement neural network into the constraint satisfaction neural network through hierarchical distillation to obtain the optimized model;

[0086] The optimization unit 26 is used to solve the constrained optimization problem using the optimization model.

[0087] The conversion unit specifically includes:

[0088] Used to introduce slack variables to transform affine constraint inequalities into affine constraint equalities.

[0089] It is used to project the original constrained optimization problem onto the affine space defined by the affine constraints, and transform it into an optimization problem with respect to the independent variables of the affine space.

[0090] The optimization model is specifically as follows: ;

[0091] in, To constrain the network The output, For precision enhancement networks The corresponding output; MAE uses the mean absolute error loss as the objective function for training the neural network; MSE uses the mean squared error loss as the objective function for training the neural network; What does it represent? What does it represent? What does it represent? What does it represent? What does it represent? What do they represent? ReLU is the activation function; KL is the divergence loss.

[0092] The first allocation unit specifically includes:

[0093] The neural network used for accuracy enhancement is pre-trained;

[0094] For use with self-supervised neural network solvers As a precision enhancement network;

[0095] The self-supervised neural network solver uses a penalty function as its loss function and controls the intensity of the penalty using training parameters.

[0096] The modeling unit includes:

[0097] This is used to simultaneously input a pre-generated dataset into both an accuracy-enhancing neural network and a constraint-satisfying neural network.

[0098] Used to then solve accuracy-enhancing neural networks and constraint-satisfying neural networks;

[0099] This is used to update the constraint-satisfying neural network using a gating function, based on the solutions of the precision-enhancing neural network and the constraint-satisfying neural network.

[0100] The principle and working process of the structure-aware dual-network collaborative optimization device have been described in detail in the section on structure-aware dual-network collaborative optimization method, and will not be repeated here.

[0101] The structure-aware module of this invention relaxes the problem by utilizing the inherent linear constraints and projects them into an affine space for decision-making, thereby transforming the original problem into a form more easily solved by neural networks. Comparative experimental results clearly show that after introducing structure awareness, constraint violations by the learning method are reduced by more than 80%. Furthermore, the accuracy gap of the DC3 method is also reduced by 7%. This is mainly because, when handling linear equality constraints, unlike the "Complete" operation in the original DC3, the structure-aware method enables the neural network to output a complete solution.

[0102] The SPDC framework proposed in this invention is used to solve constrained optimization problems with complex structures. The SPDC framework integrates three core components: problem reconstruction, knowledge extraction, and collaborative solving. Through structured guidance and model interaction, it improves solution efficiency and constraint satisfaction.

[0103] The technical solutions of the present invention will be described in detail below with reference to specific application examples. For technical details not described in the implementation process, please refer to the relevant descriptions above.

[0104] Example 1:

[0105] This invention provides a structure-aware dual-network collaborative intelligent planning method. To address the challenges related to problem structure and the critical balance between feasibility and optimality, this invention proposes a novel problem reconstruction and collaborative solution paradigm for constraint optimization. The main innovations of this invention include:

[0106] Constructing a structure-aware encoder: In the problem reconstruction phase, this invention proposes a structure-preserving strategy that analytically eliminates equality constraints through QR decomposition of the constraint matrix. This transforms the original problem into an unconstrained optimization problem on a low-dimensional manifold.

[0107] The second stage introduces an enhanced neural solver that integrates two complementary networks through knowledge sublimation: a constraint satisfaction network (CSN) responsible for ensuring feasibility, and a precision enhancement network (PEN) focused on improving the optimality of the solution. To effectively balance feasibility and optimality, this invention also designs a gated hierarchical distillation strategy that dynamically adjusts the influence of the two networks during training to mitigate potential knowledge conflicts during distillation.

[0108] A comprehensive experiment combining qp, simple nonconvex optimization, and hard nonconvex optimization validates the effectiveness of the framework: it consistently achieves zero constraint violations; under constraints, the optimality gap is reduced to 0.8% compared to the best baseline, with significant reductions in training and inference time. The results demonstrate that the SPDC framework can effectively mine problem structure, and the integrated knowledge distillation mechanism cleverly balances the optimality of the solution with constraint compliance. Notably, the framework employs fully unsupervised learning, which greatly expands its application potential in scenarios where labeled data is scarce.

[0109] This invention uses the following notation: handwritten uppercase letters (e.g., eg) () indicates a set, distribution, or loss function; bold lowercase letters (e.g., (eg, () represents a vector, and sometimes also a set of functions; while bold uppercase letters (e.g., (eg, ) represents a matrix.

[0110] Problem Statement

[0111] This invention considers the problem of minimizing an objective function under equality and inequality constraints. Here As decision variables, Let x be the problem parameters. Each x is a specific problem instance. For a given variable, the goal is to find the corresponding minimization objective function while satisfying all constraints. Theoretically, it can be nonlinear or nonconvex.

[0112] (1)

[0113] Affine constraints, in particular, are a common and important type of constraint, frequently found in problems such as resource allocation and path planning. These constraints exhibit good convexity and are easier to handle. Therefore, this invention further classifies problem constraints into affine constraints and non-affine constraints. The problem form is as follows:

[0114] Constraints (2a) and (2b) are affine constraints that may be included in the problem, where the matrix... GE Constraints (2c) and (2d) are non-affine constraints.

[0115] in, , y is

[0116] Independent variable. Therefore, the objective of this invention is to train a neural network that directly maps problem instances to their optimal solutions in an end-to-end manner. Neural networks need to predict a value that minimizes the objective function. A solution that strictly satisfies all constraints at the same time.

[0117] Self-supervised learning:

[0118] Supervised learning faces two major challenges in solving constrained optimization problems. First, obtaining high-quality supervision signals (i.e., ground truth optimal solutions) is often difficult and expensive. Second, the goal of supervised learning is to minimize the difference between predictions and provided labels, which is fundamentally different from the goal of truly learning to solve optimization problems. Therefore, supervised methods often struggle to satisfy constraints because they fail to truly capture the underlying problem structure; even small inaccuracies in prediction can propagate to cause constraint violations.

[0119] In contrast, unsupervised learning methods do not require solving instances. Instead, their goal is to learn the problem structure, including constraints and the objective function, directly from the problem statement itself. A common strategy for constrained optimization in unsupervised learning is to incorporate a penalty function into the loss function, enabling the neural network to capture both the objective and constraint structures simultaneously.

[0120] (3)

[0121] Let $\mathbf{ ... Related.

[0122] Knowledge distillation is a technique that transfers knowledge from a "teacher model" to a "student model." Its main goal is to enable the student model to replicate the teacher's reasoning ability by mimicking the teacher's output distribution or intermediate characteristics.

[0123] (4)

[0125] This represents the probability distribution obtained after scaling by temperature. Let (KL) represent the cross-entropy loss, and (KL) represent the Kullback-Leibler (KL) divergence loss. This represents the balance coefficient. The subscripts s and t correspond to the student model and teacher model, respectively.

[0126] Structure-Aware Dual-Network Collaborative Solver Framework (SPDC):

[0127] This invention introduces the core contribution of this work: the Structure-Aware Dual Network Collaboration (SPDC) framework. SPDC aims to solve constrained optimization problems through a problem-centered approach. It first reformulates the problem into a representation more suitable for neural network solvers.

[0128] Table 1 SPDC Training and Testing Process Table

[0129] Then, it employs a dual-network architecture to clearly capture information related to solution accuracy and constraint satisfaction. Finally, it utilizes knowledge distillation to effectively integrate these two aspects, providing a high-precision solution to the constrained optimization problem while ensuring strict adherence to constraints.

[0130] Figure 1 A high-level overview of SPDC is provided, illustrating its architectural design and operational logic. Given a problem instance, the process begins with structure awareness, leveraging the problem's inherent structural information to refactor it into a more suitable form.

[0131] For a neural network-based solver, the reconstructed problem is then input into the subsequent solution framework. First, a pre-trained accuracy enhancement network is obtained. In a network where constraints are satisfied During the training process, The extracted precision information is continuously integrated and refined into The trained constraint-satisfaction network integrates constraint and accuracy information. During the testing phase, the trained constraint-satisfaction network is used to solve the reconstruction problem after structure-aware processing, ultimately obtaining a high-precision solution that satisfies all constraints.

[0132] The main features of the SPDC framework can be summarized as follows:

[0133] Problem Structure Awareness: A key challenge for neural network solvers, often used as approximate differentiable optimizers, is strictly satisfying constraints in constraint optimization problems. To address this, the SPDC framework introduces the concept of problem structure awareness. This component employs a problem-driven strategy to identify and utilize potentially advantageous structural information embedded in the problem formulation. This structural insight is then used to reformulate the original problem into a form that is more effectively processed and solved by neural networks. Dual-Network Enhanced Solving Framework: Inspired by the effectiveness of knowledge distillation in integrating knowledge from different domains, this invention develops a dual-network enhanced solving framework. This architecture decomposes the complex task of constraint optimization into two more manageable sub-problems: ensuring constraint satisfaction and maximizing the accuracy of the solution. It utilizes two specialized networks: a constraint satisfaction network that captures constraint-related information, and a precision expert network focused on solution accuracy. A hierarchical knowledge distillation mechanism is then implemented to synergistically combine the knowledge learned by the two networks, ensuring that the final solution achieves both high accuracy and strict adherence to all constraints.

[0134] Supervised learning methods, particularly those based on the self-supervised problem-solving paradigm, often face challenges related to the cost and availability of labeled training data. More critically, the objective optimized during supervised training may not perfectly align with the intrinsic goal of teaching the network how to solve a problem, potentially hindering its ability to generalize or grasp underlying problem mechanisms. To overcome this, the SPDC framework operates within a self-supervised paradigm. This allows the network to learn directly from the problem's intrinsic properties, capturing fundamental information about constraints and accuracy without relying on external supervision. This approach facilitates the development of networks capable of truly learning problem-solving processes.

[0135] It is important to emphasize that SPDC defines a general framework, or a class of neural network solvers, focusing on the overall architecture rather than specifying particular implementations for its individual components. The remainder of this invention will provide more details about SPDC.

[0136] Structure SP:

[0137] Solving large-scale constrained optimization problems presents significant computational challenges. Recent research has focused on developing advanced neural network solvers to achieve fast convergence and high-quality solutions. However, a persistent challenge is balancing constraint satisfaction and solution accuracy. A key limitation of many existing methods is their inability to utilize the structural information inherent in the problem formulation. This invention introduces structure awareness, a novel approach that shifts the focus from enhancing the solver to understanding the problem structure itself. Structure awareness identifies and utilizes structural features in the problem, reformulating them into a more suitable representation that is better suited for neural network-based solvers. To illustrate the practical applicability of this approach, this invention investigates affine constraints, a class of mathematical relations widely used in mathematical optimization, computational geometry, and engineering systems. Specifically, for constrained optimization problems, constraints can be categorized into affine equality, affine inequality, non-affine equality, and non-affine inequality constraints, and are expressed as Equation (2).

[0138] First, introduce slack variables. Transform the affine inequality into an affine equality.

[0139] (5)

[0140] The relaxation problem is solved by straightforward variable substitution, specifically defined. The decision variables y and S are then re-expressed as slack variables. = , among them .

[0141] Therefore, the transformed affine equation can be combined with the original equation and expressed as:

[0142] (6)

[0143]

[0144] In the above expression

[0145]

[0146] The original problem can then be rephrased as the following formula.

[0147]

[0148] In equation (7), = , = , , are all functions of i, and the concavity and convexity of f are not limited.

[0149] According to the definition of null space, let N denote the null space of, i.e., N(·)= and 0. A feasible solution to the optimization problem must lie in the solution space of, which can be regarded as the translation of the solution space of. Therefore, the solution of can be expressed as , where is a particular solution to the original problem. To determine , Rayen et al. (Tordesillas et al., 2023) adopted a pseudoinverse method. However, this method is not suitable for ill-posed problems, requires a full-rank matrix, and has poor robustness. In contrast, the QR decomposition method computes a basis for the null space, making it highly suitable for ill-posed problems, rank-deficient matrices, and sparse scenarios. It offers greater numerical stability and robustness, avoids direct matrix inversion, and achieves higher computational efficiency.

[0150] Therefore, the present invention uses the QR decomposition method to determine the matrix and N; specifically, performing QR decomposition on can yield , where Q is an orthogonal matrix and R is an upper triangular matrix. When holds, the null space matrix is determined as , and the following formula can be derived to hold.

[0151] (8)

[0152] Based on the above derivation, the original problem is projected onto the affine space defined by affine constraints, and transformed into an optimization problem with respect to , whose expression is: (9)

[0153] Z is the independent variable of the affine space;

[0154] To ensure consistency with the form of (1), (9) is reformulated as follows: (10)

[0155] In equation (10), g(·) = g(T(·)) ) , ) = , ) g" (-) ∈ Rg are all The function. Therefore, SP senses and utilizes the affine constraints in the original data (1) and finally transforms it into the form (10), which has fewer equality constraints, lower dimensionality, and is more suitable for neural network solutions.

[0156] Precision Enhancement Network (PEN):

[0157] PEN is a network cluster designed to capture finer-grained or more precise knowledge during problem-solving without imposing strict constraints on the feasibility of the solution. The network is pre-trained to acquire detailed knowledge relevant to solving constrained optimization problems. To facilitate rapid acquisition of pre-trained models and reduce pre-training costs, this invention employs a self-supervised neural network solver. As an accuracy enhancement network, the solver uses a penalty function as its loss function and employs... (A trainable parameter) controls the strength of the penalty.

[0158] (11)

[0159] It is worth noting the penalty coefficient of the loss function. and Set to a relatively small value, allowing Be more proactive and more focused on capturing The distribution characteristics of the expert network. In a given instance, the expert network The result of reasoning It may lead to a violation of constraints, but it is more likely to retain them. More related and richer information.

[0160] The constraint satisfaction network is implemented as a network ensemble. Compared to precision-enhancing networks that emphasize solution accuracy, this ensemble is designed to prioritize rigorous feasibility, ensuring that all output solutions fully satisfy the imposed constraints—regardless of numerical optimality. This invention argues that constraint satisfaction is paramount in constrained optimization problems. Any solution that violates the constraints, no matter how precise it is with respect to the objective function, is considered infeasible and therefore lacks practical value. Therefore, the task of the constraint satisfaction network is to generate the final solution, as it has the capability to guarantee feasibility. To achieve this, this invention integrates DC3, a self-supervised learning framework developed by Donti et al., which guarantees constraint satisfaction through a dual mechanism of differentiable completion and correction. These operations aim to project the intermediate outputs of the neural network back to the feasible region, thereby ensuring that the final prediction conforms to all problem constraints. Furthermore, to improve the quality of the solution without compromising feasibility, the constraint satisfaction network is trained through a dynamic ensemble of precision-related knowledge extracted from the precision-enhancing network. This strategy enables the network to learn not only efficient but also solutions that are closer to optimal feasibility.

[0161] Constraint Satisfaction Network (CSN):

[0162] Assuming the problem is not overdetermined, for the independent variable... Training neural networks Output ITT variable entries Then according to Solving the remaining implicit equation system composed of equality constraints entries, procedures using functions This is indicated. Then, all variables are processed to satisfy the equality constraints. ,in for and . .

[0163] (12)

[0164] Since solutions derived through the equation-solving process may lack analytical differentiability, and neural networks require gradient-based optimization, the DC3 method, as a differentiable solver, mitigates this challenge by strategically implementing the implicit function theorem to maintain mathematical rigor and computational traceability (Donti et al., 2021).

[0165] DC3 introduces a gradient-based correction mechanism. via the network Adjust output This involves moving the gradient in the direction associated with the violation of the inequality constraint. The aim of this process is to guide the solution into a region where the constraint violation is less severe. Specifically...

[0166] (13)

[0167] Constraint-Saving Network It is responsible for inferring the final solution. To enhance its performance, it comes from a precision enhancement network. The precision knowledge is extracted into an auxiliary network. The auxiliary network is trained to integrate constraint-related knowledge and accuracy-related knowledge. This is achieved through an output layer distillation strategy, where... The loss function is defined as follows: (14)

[0168] set up For a given instance, the constraints satisfy the network. Output , For precision enhancement networks The corresponding output. The enhanced signal is weighted by coefficients to emphasize the importance of precision information in the distillation process.

[0169] Stratified distillation framework:

[0170] As is well known, achieving a balance between feasibility and high accuracy for approximate solutions to constrained optimization problems is inherently challenging—a compact feasibility margin is often the primary goal of approximate solutions.

[0171] network The first component of the loss function employs a soft-constraint formula commonly used in self-supervised neural solvers. However, compared to the network... The difference is that the internet Stricter requirements are imposed on constraint satisfaction. Therefore, the penalty coefficient for controlling constraint violations cannot be set too small. The second component of the loss function employs output layer distillation, where the loss comes from the network... The precision knowledge is distilled into the network This process effectively integrates constraint-related and accuracy-related knowledge. However, the inherent conflict between these two types of knowledge presents a key challenge: constraint satisfaction and accuracy improvement may generate opposing gradients during training, leading to contradictory parameter update directions. To address this issue, this invention proposes a gating mechanism combined with hierarchical knowledge distillation.

[0172] Specifically, since the completion of the operation guarantees the satisfaction of the equality constraints, a gating function is designed to evaluate the severity of violations of the inequality constraints. It takes the maximum violation of the inequality constraints in the inference result as input and outputs the corresponding distillation enhancement coefficient.

[0173] (15)

[0174] This gating function adjusts the strength of the distillation signal when the constraint violation is minimized.

[0175] The coefficient remains high, encouraging the network to learn more from accuracy experts. However, as constraint violations increase, the coefficient drops sharply—prioritizing constraint satisfaction over accuracy improvement. This adaptive mechanism ensures that the solution remains feasible while still benefiting from precision guidance where constraint satisfaction is permissible.

[0176] Experiments: The performance of the proposed SPDC framework was evaluated through a series of experiments and compared with several state-of-the-art methods reported in the literature, including DC3 and PDL (Donti et al., 2021; Park and Vanhenentryck, 2023). Experiments were conducted on convex quadratic programming, a family of simple non-convex optimization problems and a family of hard non-convex optimization problems, following the same problem settings used in previous works. All results are reported on the test set. Each experiment was run five times, and the present invention reports the average performance. All experiments were performed on a machine equipped with an NVIDIA RTX4090 (24GB) GPU, running Ubuntu 22.04, based on PyTorch 2.0.0 (Paszke et al., 2019).

[0177] Baseline approach: This invention compares SPDC with several state-of-the-art methods, including optimizing solvers, supervised learning methods, and unsupervised learning methods.

[0178] Optimizer: The benchmark's most advanced solver, considered a fundamental truth.

[0179] OSQP: A CPU-based solver for convex quadratic programming based on the alternating direction method of multipliers (Stellato et al., 2020).

[0180] qpth: A GPU-based QP solver built on the primal dual interior point method (Amos and Kolter, 2021).

[0181] IPOPT: A primitive dual interior-point method with filter line search for nonlinear optimization (Wächter and Biegler, 2006).

[0182] Supervised learning methods: Supervised learning methods use the solutions obtained from the optimizer as labels for training the neural network.

[0183] MAE: This method uses the mean absolute error (MAE) loss as the objective function for training the neural network.

[0184] MSE: This method uses mean squared error (MSE) loss as the objective function for training the neural network.

[0185] MAE+Penlty: This method uses a combination of mean absolute error (MAE) loss and constraint penalty term as the loss function for training the neural network.

[0186] MSE+Penlty: This method uses a combination of mean squared error (MSE) loss and constraint penalty term as the loss function for training the neural network.

[0187] Self-supervised learning methods: Self-supervised learning methods do not depend on the solution labels of the problem.

[0188] DC3: A self-supervised method proposed by Priya L. Donti et al., which ensures the feasibility of the results by adding differentiable operations (Donti et al., 2021).

[0189] PDL: Proposed by Seonho Park and Pascal Van henenryck, this method employs deep learning to approximate and enhance the iterative steps of the La-grangian method to identify tight problem boundaries (Park and Van henenryck, 2023). Since the original implementation was not open source, this invention faithfully replicates the method according to the paper's description to reproduce its reported performance.

[0190] "With structure awareness" means that the method includes a problem refactoring step.

[0191] Dataset:

[0192] The baseline method and the method proposed in this invention have been tested on three types of problems, including convex quadratic programming, simple non-convex optimization, and hard non-convex optimization.

[0193] Consider the following convex quadratic programming problem:

[0194] (16)

[0195] Consider the following simple nonconvex optimization:

[0196] (17)

[0197] Consider the following hard nonconvex optimization:

[0198] (18)

[0199] The Rosenbrock function, first proposed by Howard Harry Rosenbrock in 1960, is a well-known non-convex benchmark function frequently used in mathematical optimization to evaluate optimization performance.

[0200] This invention compares the performance of the SPDC method with self-supervised learning (SSL) and supervised learning (SL) methods on convex QP optimization problems. The problem instance is characterized by n=100 variables and equality constraints (…). The number of constraints (n neq) and inequality constraints (n neq) is set to 50. Red entries indicate violations of constraints, making such a solution unacceptable. The reported data is the mean of five independent trials, with the standard deviation provided in parentheses. The Optimizer method represents the baseline true values, so discrepancies are indicated with "-". Using the results in the table, Calculate the difference.

[0201] Table 2 Experimental performance data of SPDC on QP

[0202] The Rosenbrock function (Rosenbrock, 1960) poses a significant challenge to optimization methods due to its obvious non-convexity and multiple local optima. By utilizing the Rosenbrock function as the objective function for constructing a third optimization problem, this invention can rigorously evaluate the algorithm's ability to manage constraints while pursuing high solution accuracy (Iken and Lucia, 2024).

[0203] The dataset used in this invention was generated according to the same process described in DC3 (Donti et al., 2021). Specifically, the matrix... Construct a diagonal matrix, where diagonal entries are uniformly sampled from the interval [0,1] to ensure Positive semi-definite. Matrix and Sample from a standard normal distribution (0,1). Problem instance parameters. Uniform sampling is performed from the interval [−1,1].

[0204] For the three types of problems described above, various scenarios are constructed by varying the number of equality and inequality constraints. Specifically, each problem is evaluated at nine different scales. The number of equality constraints is fixed at 50, while the number of inequality constraints is set to 10, 30, 50, 70, and 90. Alternatively, the number of inequality constraints is fixed at 50, while the dimensions of the equality constraints vary between 10, 30, 50, 70, and 90. For each scale configuration, 10,000 instances are generated. The dataset is divided into training, validation, and test sets in a 10:1:1 ratio.

[0205] Evaluation indicators

[0206] This invention evaluates the SPDC framework in terms of constraint satisfaction, solution optimality, and computational efficiency, and compares its performance with other state-of-the-art methods.

[0207] Feasibility: Measures the degree to which the framework satisfies the problem constraints. This is evaluated using the maximum and average violations of equality and inequality constraints, denoted as the metrics Max.eq, Mean.eq, Max.ineq, and Mean.ineq.

[0208] Optimality: Evaluates the quality of the solution based on objective values, aiming to demonstrate the accuracy of the framework results, and is represented as a metric object.val.

[0209] Speed: Evaluate the solution speed of the framework, including training and inference time.

[0210] Table 3. Experimental evaluation of the performance of convex quadratic programming at different scales.

[0211] Table 4 shows the time results for convex quadratic programming. The problem dataset contains nine different constraint scales. "Instance generation time" refers to the time spent generating the corresponding instance for each scale. When using the structure-aware module, this component specifically covers the time for instance generation and constraint transformation. "Total time" measures the complete duration from the initial problem generation to the completion of the solution.

[0212] Table 4. Time Results of Convex Quadratic Programming

[0213] Table 5 shows the hyperparameters of the supervised learning method. The hyperparameters of DC3 are consistent with the optimal parameters proposed in the DC3 paper, and the hyperparameters of PDL are consistent with the optimal parameters proposed in the PDL paper. This invention conducted simple tests on these hyperparameters and finally selected the final bolded hyperparameters in the table.

[0214] Table 5 Hyperparameter Table for Algorithm Use

[0215] Table 6. Experimental Evaluation Table for Non-convex Optimization

[0216] Parameter settings:

[0217] The experiments for DC3 and PDL followed the setup of the original paper. Except for PDL, all neural network models in this study employed two hidden layers, each with 200 units, using the ReLU activation function. To ensure experimental fairness, the inner loop iteration count for PDL was set to 50, thus ensuring a consistent total training iteration count of 1000 for all learning methods. The parameter configurations of the proposed methods are comprehensively recorded in Table 4.

[0218] Experimental evaluation:

[0219] This invention provides a rigorous experimental evaluation of the SPDC framework. The effectiveness of the algorithm was tested on optimization problems with three different constraint scales: QPs, simple nonconvex optimization, and difficult nonconvex optimization. Then, the performance of this framework was systematically compared with established baseline methods (including supervised learning models, self-supervised learning algorithms, and conventional optimization solvers).

[0220] To facilitate clear interpretation of the research results, the computational results for problem instances characterized by 50 equality constraints and 50 inequality constraints are provided separately and in detail for each problem category. Subsequently, test results from instances with optional constraint numbers within the same category are presented in a uniform format.

[0221] As shown in Table 1, among the baseline methods without structure awareness, only the DC3 method can guarantee the feasibility of the results, while other learning methods exhibit varying degrees of constraint violations. This invention reproduces the performance of the PDL method. To ensure the fairness of the results, this invention only adjusts its iterations to 1000. Experimental results show that although the PDL algorithm has a strong ability to find tight boundaries, it cannot satisfy all constraints. Although the DC3 method satisfies the constraints, its accuracy is significantly lower, reaching 10.58%. This is mainly because: while the "Complete" operation can complete the remaining variables according to the equality constraints, if it is in the output part of the neural network, the remaining variables completed by this operation will inevitably introduce a bias; and the "Correct" operation only performs gradient descent correction on the part that violates the inequality constraints, aiming to reduce the degree of violation without fully considering accuracy. Therefore, the "Complete" and "Correct" operations, to some extent, undermine the optimality of the DC3 method. The observation results clearly show that all learning methods violate equality constraints significantly more often than inequality constraints. The fundamental reason is that equality constraints impose strict requirements on the accuracy of the solution, which fundamentally conflicts with the inherent characteristic of neural networks as approximate solutions. Therefore, neural networks struggle to effectively learn and fully satisfy equality constraints. The performance of neural networks is closely related to the inherent structure of the problem. Many current methods fail to fully utilize the structural information of the problem, instead treating easily handled structures like affine constraints the same as complex structures and generalizing them, which in turn weakens the potential of neural networks. The structure-aware module proposed in this invention relaxes the problem by utilizing the inherent linear constraints and projects them into an affine space for decision-making, thereby transforming the original problem into a form more easily solved by neural networks. Comparative experimental results clearly show that after introducing structure awareness, constraint violations by the learning method are reduced by more than 80%. Furthermore, the accuracy gap of the DC3 method is also reduced by 7%. This is mainly because, when handling linear equality constraints, unlike the "Complete" operation in the original DC3, the structure-aware method enables the neural network to output a complete solution. Figure 3 As shown, since the output of the neural network is relatively independent, errors in some outputs do not affect the accuracy of the remaining solutions in the chain reaction, thus improving the overall solution accuracy of DC3. Nevertheless, currently only the DC3 method can consistently satisfy all constraints after introducing structure awareness, and its accuracy performance still has room for further optimization.

[0222] Table 7. Experimental evaluation of the performance of simple nonconvex optimization at different scales.

[0223] Table 8 Time Results of Simple Non-convex Optimization The core of the SPDC framework is a "precision expert network" built on a neural network, using a simple penalty function as its loss function. With the help of a "hierarchical distillation" technique, SPDC effectively overcomes the potential conflict between the precision gradient and the constraint gradient, achieving strict satisfaction of all constraints. Experimental results show that SPDC's precision gap is only 0.804, 9.7% lower than DC3's gap, while its inference efficiency is 169 times that of DC3. Table 2 summarizes the results of other examples, which are highly consistent with the above findings. Figure 4 It provides a more intuitive visualization.

[0224] Table 2 fully demonstrates the experimental performance of SPDC on qp, further proving that it can satisfy the constraints and achieve better accuracy in all test cases. It is worth noting that... Figure 2 In the diagram, the gray data points representing the SPDC results are mainly concentrated in the target area in the lower left corner, indicating good performance.

[0225] Table 4 summarizes the solution and training times of the QP. Data shows that the structure-aware module's runtime is only 0.01 to 0.02 seconds, its time overhead is almost negligible. In terms of training efficiency, SPDC is 4 times faster than DC3. More importantly, even considering the pre-training time of the Precise Expert Network (PEN), SPDC's total training time is still 3 times faster than DC3.

[0226] Applying SPDC to qp is insufficient to fully demonstrate its potential in handling complex optimization tasks. Therefore, this invention further tests it on a simple nonconvex optimization problem, and the results are detailed in Table 6.

[0227] Further observation of the original problem revealed that only DC3 consistently satisfied all constraints, while other learning-based methods violated equality constraints significantly more often than inequality constraints. A significant reduction in constraint violations was observed in these learning methods after integrating structure awareness. Notably, DC3 achieved a 5.5% improvement in accuracy under structure awareness, but it remains the only method that fully satisfied the constraints, suggesting potential for further accuracy improvement. Finally, through hierarchical distillation training, the SPDC framework of this invention achieved complete constraint satisfaction and improved accuracy to an optimality gap of 0.682%. This represents a 7.18% improvement in accuracy and a 70-fold increase in inference speed compared to DC3.

[0228] In contrast, a comparison before and after applying the structure-aware module in the analysis of qp yielded an interesting insight. While the transformation reduced the number of constraints and the dimensionality of the problem, the solver's solution time paradoxically increased. This phenomenon is likely to occur because the non-convex characteristics of the problem become more apparent or amplified after the transformation, thus complicating the search for the optimal solution and posing a greater challenge to traditional solvers. This suggests that while this transformation strategy is advantageous for neural network solvers, it may not be universally applicable to all solver types, especially on non-convex instances. Table 7 lists experimental results for other examples with different constraint scales, further confirming the above viewpoint.

[0229] The solutions and training times of the experiments in this invention are summarized in Table 8. It can be observed that the structure-aware module takes only about 0.03 seconds, with almost no additional overhead. The training speed of SPDC is twice as fast as DC3, and even considering the pre-training time of PEN, its total training time is still shorter than that of DC3.

[0230] In addition to the two types of problems discussed above, this invention further tests and evaluates its method on the more significant nonconvexity optimization problem of hard nonconvexity. The results are shown in Table 9.

[0231] In the original hard nonconvex optimization environment, even though DC3 typically exhibits strong constraint satisfaction, it still produces slight constraint violations. However, after integrating the structure-aware module, a reduction in constraint violations was observed in all learning methods, similar to the findings on simpler problems. Notably, DC3 not only fully satisfies the constraints but also improves accuracy by 1.96%.

[0232] The SPDC framework proposed in this invention maintains the feasibility of the solution for this challenging non-convex optimization problem, achieving an impressive optimality gap of only 0.381%. Compared to structure-aware DC3, this translates to a 2.444% improvement in accuracy and a 30x increase in inference speed. Consistent with observations in other non-convex scenarios, this problem transformation, while beneficial to learning-based solvers, is less favorable to traditional optimal solvers, leading to increased solution time. Results for the remaining problem instances, as shown in Table 10, further support these conclusions.

[0233] Table 11 summarizes the solutions and training times of the experiments of this invention. Consistent with the observations of the first two types of problems, the running time of the structure-aware module is only about 0.02 seconds, and its time overhead is almost negligible. In terms of training efficiency, SPDC's training speed is twice that of DC3; even including the pre-training time of PEN, the total training time of SPDC is still significantly shorter than that of DC3.

[0234] Table 9 Experimental evaluation of hard nonconvex optimization

[0235] Table 10. Experimental Evaluation of Hard Non-convex Optimization at Various Scales

[0236] Table 11 Time Results of Hard Non-convex Optimization

[0237] This invention proposes a novel framework, SPDC, to solve constrained optimization problems with complex structures. The SPDC framework integrates three core components: problem reconstruction, knowledge extraction, and collaborative solving. Through structured guidance and model interaction, it improves solution efficiency and constraint satisfaction.

[0238] The structure-aware module of this invention relaxes the problem by utilizing the inherent linear constraints and projects them into an affine space for decision-making, thereby transforming the original problem into a form more easily solved by neural networks. Comparative experimental results clearly show that after introducing structure awareness, constraint violations by the learning method are reduced by more than 80%. Furthermore, the accuracy gap of the DC3 method is also reduced by 7%. This is mainly because, when handling linear equality constraints, unlike the "Complete" operation in the original DC3, the structure-aware method enables the neural network to output a complete solution.

[0239] The SPDC framework proposed in this invention is used to solve constrained optimization problems with complex structures. The SPDC framework integrates three core components: problem reconstruction, knowledge extraction, and collaborative solving. Through structured guidance and model interaction, it improves solution efficiency and constraint satisfaction.

[0240] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A structure-aware dual-network collaborative intelligent planning method, characterized in that, The method includes the following steps: Transform the constrained optimization problem; The transformed constrained optimization problem is decomposed into a constraint problem and an accuracy problem; Assign the accuracy problem to an accuracy-enhancing neural network; Assign the constraint problem to a constraint satisfaction neural network; The precision enhancement neural network is distilled into the constraint satisfaction neural network through hierarchical distillation to obtain the optimized model; The constrained optimization problem is solved using the optimization model described above.

2. The structure-aware dual-network collaborative intelligent planning method according to claim 1, characterized in that, The transformation of the constrained optimization problem specifically includes: By introducing slack variables, the affine constraint inequality is transformed into an affine constraint equality. The original constrained optimization problem is projected onto the affine space defined by the affine constraints and transformed into an optimization problem with respect to the independent variables of the affine space.

3. The structure-aware dual-network collaborative intelligent planning method according to claim 1, characterized in that, The optimization model is specifically as follows: ； in, To constrain the network The output, For precision enhancement networks The corresponding output; MAE uses the mean absolute error loss as the objective function for training the neural network; MSE uses the mean squared error loss as the objective function for training the neural network; What does it represent? What does it represent? What does it represent? What does it represent? What does it represent? What do they represent? ReLU is the activation function; KL is the divergence loss.

4. The structure-aware dual-network collaborative intelligent planning method according to claim 1, characterized in that, The assignment of accuracy issues to accuracy-enhancing neural networks specifically includes: The precision-enhancing neural network is pre-trained; Self-supervised neural network solver As a precision enhancement network; The self-supervised neural network solver uses a penalty function as its loss function and controls the intensity of the penalty using training parameters.

5. The structure-aware dual-network collaborative intelligent planning method according to claim 1, characterized in that, The process of distilling the precision-enhancing neural network into a constraint-satisfying neural network through hierarchical distillation to obtain an optimized model includes: The pre-generated dataset is simultaneously input into both the accuracy-enhancing neural network and the constraint-satisfying neural network; Then, solve for the precision-enhancing neural network and the constraint-satisfying neural network; Based on the solutions of the precision enhancement neural network and the constraint satisfaction neural network, the constraint satisfaction neural network is updated using a gating function.

6. A structure-aware dual-network collaborative intelligent planning device, characterized in that, include: The transformation unit is used to transform constrained optimization problems. Decomposition units are used to decompose the transformed constrained optimization problem into a constraint problem and a precision problem; The first allocation unit is used to assign accuracy problems to the accuracy enhancement neural network; The second allocation unit is used to allocate the constraint problem to the constraint satisfaction neural network; The modeling unit is used to distill the accuracy-enhancing neural network into a constraint-satisfying neural network through hierarchical distillation to obtain an optimized model. An optimization unit is used to solve the constrained optimization problem using the optimization model.

7. The structure-aware dual-network collaborative intelligent planning device according to claim 6, characterized in that, The conversion unit specifically includes: Used to introduce slack variables to transform affine constraint inequalities into affine constraint equalities. It is used to project the original constrained optimization problem onto the affine space defined by the affine constraints, and transform it into an optimization problem with respect to the independent variables of the affine space.

8. The structure-aware dual-network collaborative intelligent planning device according to claim 6, characterized in that, The optimization model is specifically as follows: ； in, To constrain the network The output, For precision enhancement networks The corresponding output; MAE uses the mean absolute error loss as the objective function for training the neural network; MSE uses the mean squared error loss as the objective function for training the neural network; What does it represent? What does it represent? What does it represent? What does it represent? What does it represent? What do they represent? ReLU is the activation function; KL is the divergence loss.

9. The structure-aware dual-network collaborative intelligent planning device according to claim 6, characterized in that, The first allocation unit specifically includes: The neural network used for accuracy enhancement is pre-trained; For use with self-supervised neural network solvers As a precision enhancement network; The self-supervised neural network solver uses a penalty function as its loss function and controls the intensity of the penalty using training parameters.

10. A structure-aware dual-network collaborative intelligent planning device according to claim 6, characterized in that, The modeling unit includes: This is used to simultaneously input a pre-generated dataset into both an accuracy-enhancing neural network and a constraint-satisfying neural network. Used to then solve accuracy-enhancing neural networks and constraint-satisfying neural networks; This is used to update the constraint-satisfying neural network using a gating function, based on the solutions of the precision-enhancing neural network and the constraint-satisfying neural network.

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