Chaotic quantum enhanced snow goose algorithm for complex optimization problems

Abstract

This invention discloses a chaotic quantum-enhanced Snow Goose algorithm for complex optimization problems. This algorithm combines a chaotic mapping mechanism with a quantum rotation gate strategy, aiming to achieve a more optimized trade-off between the exploration and development phases. First, a nonlinear chaotic system is introduced, utilizing its inherent randomness and ergodicity to generate dynamic perturbations, thereby enhancing population diversity in the search phase and preventing premature convergence. Second, based on the quantum superposition principle, a quantum rotation gate operation is used. Leveraging the characteristics of quantum superposition states, a strategy is designed to periodically update individuals, guiding the population towards the optimal region for precise development, significantly improving convergence accuracy. Although the quantum rotation gate primarily performs local fine-tuning of individuals, it guides them to the most globally recognized positions, indirectly promoting global search. The chaotic quantum-enhanced Snow Goose algorithm effectively optimizes performance by embedding these two strategies into the main loop of the SGA and executing them in alternating periodic phases.

Description

Technical Field

[0001] This invention relates to a chaotic quantum-enhanced Xueyan algorithm for complex optimization problems, belonging to the field of computer artificial intelligence technology. Background Technology

[0002] In recent years, heuristic algorithms have attracted widespread attention due to their superior ability to solve complex optimization problems. The Snow Goose Algorithm (SGA), as a novel heuristic algorithm, has demonstrated excellent performance in solving spherical function optimization problems, thus proving its effectiveness in this field. However, its efficiency decreases when faced with complex problems because the Snow Goose Algorithm (SGA) struggles to maintain its global exploration scope and refine local development accuracy, thereby limiting the accuracy of its solutions.

[0003] Heuristic algorithms have become indispensable tools for solving complex optimization challenges that traditional mathematical methods struggle with. Drawing inspiration from biological behavior, physical rules, or natural phenomena, these algorithms provide a robust and flexible framework for finding satisfactory solutions in fields such as aerodynamic airfoil design, civil engineering truss structure optimization, and advanced composite material design. The main appeal of these methods lies in their robustness in exploring complex search spaces, often discovering efficient solutions to complex optimization problems where traditional methods (such as gradient-based optimizers) struggle to find the global optimum. This fundamental appeal has spawned a wide variety of heuristic algorithms, broadly categorized according to their basic sources of inspiration. Evolutionary algorithms (EAs), such as genetic algorithms, simulate biological evolution through operations like selection, crossover, and mutation. Physics-based algorithms, such as simulated annealing and gravity search algorithms, are based on principles of physical processes. Swarm intelligence (SI) algorithms, including particle swarm optimization and ant colony optimization, mimic the collective and decentralized intelligence of social organisms such as birds, ants, or fish. Among these diverse optimizers, the Snow Goose Algorithm (SGA) represents a relatively novel and interesting algorithm that mimics the unique migration patterns of snow goose flocks. Its core innovation lies in the structural distinction between two flight modes: an energy-efficient "V" formation for cooperative and extensive exploration of the search space, and a simple "straight line" formation for rapid and targeted exploration of the best discovery areas.

[0004] The Snow Goose Algorithm (SGA) has established its ability to address a range of optimization challenges. Recognizing the algorithm's potential for improvement, contemporary research has strived to refine its fundamental mechanisms using various approaches. For example, to enhance population diversity and mitigate premature convergence, B. Zhao et al. combined adaptive switching with stochastic mechanisms (ESGA), while Bian et al. introduced biomimetic mechanisms such as leader rotation and horn-guided (ISGA). Elsewhere, hybrid approaches (RLSGA) with other computational intelligence techniques, such as deep reinforcement learning, have been explored to enhance local search capabilities and adaptability. Applications have also expanded to complex domains, such as grid optimization and hyperparameter tuning, demonstrating the practicality of SGA. Despite these valuable and consistent efforts, a rigorous review reveals two persistent core challenges. The first is the inherent difficulty in achieving a robust and dynamic balance between global exploration and local development. Most existing modifications, while achieving performance improvements in specific scenarios, still rely on rule-based tuning or include additional stochastic components. These approaches often equate to a more complex form of trial and error, lacking a theory-based mechanism that adapts to the exploration-exploration process throughout the search, leveraging trade-offs. The second challenge lies in enhancing the ability of algorithms to reliably solve complex real-world problems governed by fundamental physical laws, where the accuracy and convergence robustness of the solutions are crucial. This fundamental limitation hinders performance and versatility across diverse and computationally demanding problem domains.

[0005] The Snow Goose Algorithm (SGA) is a nature-inspired metaheuristic optimizer that mimics the migratory behavior of snow goose flocks. Its core innovation lies in structurally distinguishing between two flight modes: an energy-efficient "V" formation for cooperative and extensive exploration of the search space, and a simple "straight line" formation for rapid and targeted exploration of optimal discovery areas.

[0006] In recent years, SGA has made significant progress through applications in various fields. It has been used for hyperparameter optimization in gesture recognition deep learning models, achieving extremely high accuracy; for parameter optimization in bearing fault diagnosis; for power grid optimization; for multi-objective logistics planning using surrogate models (GP-SAMOSGA); and for solving complex economic dispatching problems.

[0007] To address the inherent limitations of the algorithm, various improvements to the core SGA have been proposed. For example, to mitigate sensitivity to initial conditions and enhance diversity, B. Zhao et al. proposed an ESGA combining adaptive switching and stochastic policies, while Bian et al. introduced biomimetic mechanisms such as leader rotation and horn guidance in their ISGA. Elsewhere, hybridization with other computational intelligence techniques has been explored to improve local search capabilities and convergence accuracy. Notably, Liu et al. fused SGA with deep reinforcement learning (RLSGA) to guide agent behavior in robot calibration, and J. Wu et al. employed a surrogate model (GP-SAMOSGA) for complex logistics planning. Meanwhile, Fu et al. set a high-performance benchmark with their novel RBMO algorithm, against which subsequent optimizers are frequently compared.

[0008] Despite these advancements, SGA and its variants still face several ongoing challenges. The algorithm exhibits significant sensitivity to initial conditions, where suboptimal population initialization can severely impact performance. Furthermore, its fundamental structure lacks a sophisticated local search mechanism, limiting accuracy in complex search spaces. Existing improvements have also introduced new problems: adaptive strategies often rely on simplistic parameter tuning, heuristic diversity mechanisms lack mathematical foundations, and hybrid methods significantly increase computational complexity. The fundamental challenge remains achieving an effective balance between global exploration and local development.

[0009] Significant improvements and applications have been made in this research. Leung et al. proposed PINN based on neural homogenization (NH-PINN), a three-step method that utilizes mathematical homogenization theory to solve multi-scale problems, significantly improving accuracy. Qiu et al. introduced coupled automatic numerical differentiation PINN (can-PINN), which unifies automatic and numerical differentiation to compute the loss function, enabling robust training with sparse data. Meng et al. combined PINN with first-order reliability methods to create PINN-FORM, a black-box solver for structural reliability analysis using implicit partial differential equations. In terms of applications, Raissi et al. employed multiple coupled deep neural networks to solve challenging inverse fluid-structure interaction problems, accurately inferring parameters from minimal data. Cai et al.'s comprehensive review further demonstrates the effectiveness of PINN for inverse problems in various complex flows. While these works are ingenious in solving specific challenges (such as multi-scale features, derivative computation, and inverse problems), they generally suffer from a key standard limitation: a significant neglect of the neural architecture search (NAS) inherent in PINN itself and the improvements proposed by Leung et al. Meng et al. and Raissi et al. primarily focused on algorithms and mathematics, emphasizing preprocessing, hybrid loss definitions, or multi-network coupling. They operated under the implicit assumption that standard (often arbitrarily chosen) fully connected network architectures were sufficient. Similarly, Chiu et al.'s can-PINN focused on loss computation without questioning the optimality of the architecture. Cai et al.'s comments also failed to consider architecture choice as a critical bottleneck. This oversight is significant because suboptimal architectures, regardless of the complexity of the training strategy or loss function, severely hinder a network's ability to learn complex physical phenomena. Consequently, these methods often require inefficient manual hyperparameter tuning and may fail to achieve the full potential accuracy. This work addresses this significant gap by introducing a neural architecture search approach to PINN, which automatically optimizes the network structure to better conform to the laws of physics. Summary of the Invention

[0010] The purpose of this invention is to address the shortcomings and deficiencies of existing technologies by proposing a chaotic quantum-enhanced Snow Goose algorithm for complex optimization problems. This algorithm combines a chaotic mapping mechanism with a quantum rotation gate strategy, aiming to achieve a more optimized trade-off between the exploration and development phases. First, a nonlinear chaotic system is introduced, utilizing its inherent randomness and ergodicity to generate dynamic perturbations, thereby enhancing population diversity in the search phase and preventing premature convergence. Second, based on the quantum superposition principle, a quantum rotation gate operation is used. Leveraging the characteristics of quantum superposition states, a strategy for periodically updating individuals is designed to guide the population towards the optimal region for precise development, significantly improving convergence accuracy. Although the quantum rotation gate primarily performs local fine-tuning of individuals, it guides them to the most globally recognized positions, indirectly promoting global search. The chaotic quantum-enhanced Snow Goose algorithm effectively optimizes its performance by embedding these two strategies into the main loop of the SGA and executing them in alternating periodic phases. These two mechanisms work together in this periodic interactive manner to balance the algorithm's global exploration and local development capabilities. To verify the effectiveness of the proposed algorithm, this invention further applies the chaotic quantum-enhanced Snow Goose algorithm to the automatic architecture search of Physical Information Neural Networks (PINNs) to solve complex partial differential equations (PDEs). Extensive experiments on various benchmark functions demonstrate that the chaotic quantum-enhanced Snow Goose algorithm achieves superior performance in terms of convergence speed and solution accuracy. Furthermore, in practical PINN applications, it exhibits enhanced robustness and effectiveness, confirming its overall superiority.

[0011] The technical solution adopted by this invention to solve its technical problem is: a chaotic quantum-enhanced snow goose algorithm for complex optimization problems, which includes the following steps: Step 1: Initialize the snow goose flock. A chaotic mapping method is used to randomly generate the initial position and velocity of each snow goose in the solution space. This ensures that the population distribution in the search space has good ergonomics and diversity, avoiding aggregation problems that may arise from traditional random initialization.

[0012] Step 2: Calculate the fitness value of each snow goose in the population. Based on the type of optimization problem, compare the fitness of all snow geese, find the snow goose with the best fitness in the current population, and record its position as the global optimum.

[0013] Step 3: Check if the algorithm has reached the preset termination condition. If it is met, the algorithm ends and outputs the global optimal solution; if not, enter the main loop and continue to execute Step 4.

[0014] Step 4: Based on the core formula of the Chaotic Quantum Enhanced Snow Goose Algorithm, and combining the snow goose's own historical best position with the population's global best position, update the flight speed and position of each snow goose.

[0015] Step 5: This step introduces a multi-mode development mechanism based on snow goose flight formations. It determines whether a "V-formation" strategy is currently being used. If it is a V-formation, the V-formation development strategy is executed. In this mode, snow geese primarily utilize information surrounding the global optimum for in-depth exploration, and there may be a leader-follower relationship among the snow geese, similar to that in a V-formation, emphasizing local development. If it is not a V-formation, a V-shaped development strategy is executed. This mode may simulate another group collaboration model, balancing the exploration of individual optimal experience and group experience by snow geese, or focusing on exploration and development in different directions.

[0016] Step 6: Perform boundary checks on the updated snow goose positions. If a snow goose position exceeds the preset solution space boundary, a specific boundary handling strategy is used to correct it to the legal range, ensuring that the search is conducted within the feasible region.

[0017] Step 7: Regardless of which development mode was used in Step 5, a linear development strategy will be executed thereafter. This mode simulates the straight-line flight of a flock of geese, guiding the snow geese to search more directly and efficiently along the current optimal direction, further enhancing the algorithm's convergence ability.

[0018] Step 8: Recalculate the fitness values ​​of all snow geese after updating their positions in Steps 4-7. Update the historical best position for each individual snow goose. Compare and update the global best position and corresponding best fitness value for the entire population.

[0019] Step 9: Perform a quantum rotation operation once every 3 iterations. If the quantum rotation operation is not performed, proceed to step 10. This operation draws on the concept of quantum computing, updating the snow goose's position through a specific "rotation," possibly aiming to help the snow goose escape local optima, increase population diversity, and enhance global exploration capabilities.

[0020] Step 10: Set the iteration counter The algorithm then jumps back to step 3 and begins a new round of iterations until the termination condition is met.

[0021] Beneficial effects: 1. This invention proposes a chaotic quantum-enhanced snow goose algorithm for complex optimization problems, which can effectively balance the global exploration and local exploitation of the enhanced optimizer.

[0022] 2. This invention can effectively combine chaotic mapping to enhance population diversity and quantum rotation gate to guide the population toward the optimal solution.

[0023] 3. This invention demonstrates the superiority of CQ-SGA over original and advanced algorithms on the CEC 2017 / 2022 benchmark tests. 4. This invention discretizes the proposed chaotic quantum-enhanced snow goose algorithm and applies it to PINN architecture search, resulting in models with higher accuracy and robustness. Attached Figure Description

[0024] Figure 1 This is a flowchart of the algorithm of the present invention.

[0025] Figure 2 This is a flowchart of the CQ-SGA workflow for PINN.

[0026] Figure 3 Convergence curves of CQ-SGA and the original algorithm on a specific CEC 2017(30D) function: (a) F1, (b) F5, (c) F18, (d) F29.

[0027] Figure 4 Convergence curves of CQ-SGA and the original algorithm on a specific CEC 2022(10D) function: (a) F1, (b) F2, (c) F7, (d) F11.

[0028] Figure 5 Convergence curves of CQ-SGA and advanced algorithms on specific CEC 2017(30D) functions: (a) F2, (b) F5, (c) F12, (d) F23.

[0029] Figure 6 Convergence curves of CQ-SGA and advanced algorithms on specific CEC 2022(10D) functions: (a) F1, (b) F3, (c) F6, (d) F12.

[0030] Figure 7 Convergence curves of CQ-SGA and the original algorithm on a specific CEC 2017(50D) function: (a) F1, (b) F5, (c) F18, (d) F29.

[0031] Figure 8 Convergence curves of CQ-SGA and the original algorithm on a specific CEC 2022(20D) function: (a) F1, (b) F2, (c) F7, (d) F11.

[0032] Figure 9Convergence curves of CQ-SGA and advanced algorithms on specific CEC 2017(50D) functions: (a) F3, (b) F5, (c) F11, (d) F25.

[0033] Figure 10 Convergence curves of CQ-SGA and advanced algorithms on specific CEC 2022(20D) functions: (a) F1, (b) F3, (c) F6, (d) F12.

[0034] Figure 11 This is a schematic diagram showing the average error values ​​of the SGA variants at CEC 2017 and CEC 2022.

[0035] Figure 12 This diagram illustrates the classification of SGA variant results for CEC 2017 and CEC 2022.

[0036] Figure 13 For N chaos and A schematic diagram of the sensitivity analysis.

[0037] Figure 14 This is a schematic diagram of the loss curves for fine-grained search and coarse-grained search in the PINN architecture.

[0038] Figure 15 A schematic diagram comparing the solution domains of the Burgers equation: true solution (a), CQ-SGA reconstruction (b), and PSO reconstruction (c). Detailed Implementation

[0039] The invention will now be described in further detail with reference to the accompanying drawings.

[0040] like Figure 1As shown, this invention provides a chaotic quantum-enhanced Snow Goose (SGA) algorithm for complex optimization problems. This algorithm integrates chaotic mapping and quantum rotation gate strategies to directly address the core challenge of balancing global exploration and local exploitation. First, chaotic mapping is introduced to generate the initial population. Its ergodicity and randomness ensure excellent diversity, laying a solid foundation for global exploration. Second, the quantum rotation gate mechanism guides individual position updates. By utilizing the principle of quantum superposition, it can guide local search probabilities towards elite solutions, replacing random wandering with intelligent, targeted exploitation. These strategies are embedded in the main loop of the SGA, working collaboratively and periodically, seamlessly integrating with its inherent biphasic structure. Chaotic mapping is applied in the "V-shape" formation phase of the SGA to support exploration, and quantum rotation gates are activated in the "straight line" formation phase of the SGA to refine exploration, ensuring a dynamic balance between exploring new regions and refining promising regions. To verify the performance of CQ-SGA in complex scenarios, this invention was experimentally tested across standard benchmark functions and applied to the automated architecture search of PINN, a problem involving high-dimensional and discrete search spaces. Extensive experiments on various benchmark functions demonstrate that CQ-SGA achieves superior performance in solution accuracy. In practical applications with PINN, it also exhibits improved robustness and efficiency, confirming its overall superiority and practicality.

[0041] PINN has become a powerful paradigm for solving partial differential equations by directly encoding the governing physical laws into the loss function of a neural network. This inherent ability to seamlessly integrate data and mathematical models has sparked widespread research interest, leading to numerous enhancements aimed at improving its accuracy and applicability across various scientific fields.

[0042] This invention provides a comprehensive exposition of the proposed CQ-SGA. First, it briefly reviews the core principles of the original SGA, which form the basis for the enhanced functionality of this invention. Then, it details two key mechanisms: the introduction of chaotic mappings to improve initialization quality and enable compelling global exploration, and quantum rotation gate operations for improving local exploitation of the algorithm. Next, these two mechanisms are intricately integrated with the SGA algorithm, presenting the complete steps of the proposed CQ-SGA. Finally, this invention applies CQ-SGA to a practical scenario: architecture search for PINN.

[0043] This invention briefly outlines the original SGA

[25] , a metaheuristic algorithm derived from natural phenomena that simulates the long-distance migration habits of snow geese. The algorithm optimizes the search by simulating two energy-efficient flight formations adopted by the flock: a "V" formation and a "straight" formation. These behavioral patterns adaptively switch based on dynamic angle parameters associated with iteration counts. The algorithm first generates the position and velocity of the snow goose population within the search space. The fitness value of each candidate solution goose is evaluated to determine the initial global optimal solution. During optimization, the algorithm switches between the two modes based on the phase angle. In the "V" exploration phase, a hierarchical update strategy based on fitness ranking is adopted: the top 20% of individuals move directly to the global leader, the middle 60% are influenced by both the leader and the population center, and the bottom 20% are additionally added to the avoidance mechanism towards the worst solution, thereby enhancing diversity and preventing premature convergence. After entering the "straight" utilization phase, the algorithm randomly selects one of two strategies with a 50% probability: one simulates individuals following randomly selected stronger companions, and the other simulates individuals following randomly selected stronger companions.

[0044] Another approach introduces Brownian motion (a type of random walk), which applies random perturbation to the current position, helping the algorithm escape local optima.

[0045] While the design of the SGA is commendable, it still faces challenges in maintaining a robust balance between search strategies. To address this fundamental limitation, this invention introduces two key mechanisms: chaotic mapping and quantum rotation gates.

[0046] The original SGA suffers from poor initialization quality and insufficient global exploration capability. To overcome these limitations, this invention introduces a chaotic mapping strategy based on an exponential discrete memristor (EDM) system. This mechanism enhances the population initialization process and global exploration capability by replacing traditional random generation with a deterministic chaotic sequence. It transforms initialization from a simple random process into a systematic exploration of the search space through traversing chaotic dynamics. The core principle involves generating a chaotic sequence using Equation 1: This represents the updated state variable.

[0047] For population initialization, the chaotic sequence is mapped to the actual search space through a linear transformation, as shown in Equation 2: The final value after the generation.

[0048] During the optimization process, chaotic perturbations are dynamically injected to maintain diversity, as shown in Equation 3: The vector can create different perturbation directions in different dimensions.

[0049] The chaotic mapping mechanism effectively improves the initialization quality and global exploration capability of the SGA. It ensures full coverage of the search space by traversing the chaotic sequence, provides diversity maintenance of the system throughout the optimization process, prevents premature convergence, and enhances the performance of the algorithm in complex optimization environments.

[0050] The original SGA exhibits inherent limitations in its local development capabilities. To overcome this shortcoming, this invention introduces a quantum rotation gate strategy. This mechanism refines the local search behavior into precise, quantum-inspired operations by replacing random updates with a series of updates. It transforms the solution update process from random perturbation to guided rotation adjustments targeting elite regions.

[0051] The core principle involves representing the position of a single solution as a qubit in Hilbert space, as defined in Equation 4: The rotating door dynamically adjusts the quantum state through unitary transformation, as shown in Equation 5: The quantum rotation gate mechanism, by combining the principle of quantum superposition, effectively improves the local search accuracy of SGA. It supports fine-grained directional updates, significantly improving convergence accuracy while retaining the global search capability of the original algorithm. This invention will now fully describe the complete CQ-SGA workflow. CQ-SGA is an enhanced nature-inspired metaheuristic optimization algorithm built upon SGA, integrating chaotic mapping and QRG mechanisms. It mimics the migratory behavior of snow geese while incorporating mathematical innovations to enhance optimization performance. The algorithm operates through three core stages: chaotic initialization of the snow goose's position and velocity, dynamic position update with chaotic perturbations, and quantum-inspired refinement through rotation transformation. The complete algorithm workflow is summarized in Algorithm 1. The CQ-SGA flowchart is shown below. Figure 1 It is shown intuitively. The algorithm uses chaotic mapping to generate and maintain the positions P and velocities V of the population, replacing the random initialization of the original SGA, as shown in Equation 7: In the V-shaped phase, a V-shaped formation was adopted, where position updates were enhanced through chaotic perturbations and stratified according to fitness levels. The update rules for different levels are as follows: the top 20% follow the global leader with additional chaotic guidance; the middle 60% balance the attraction to the leader and the repulsion from the group center through chaotic crossover operations; and the weakest 20% are added to the velocity-based update chaotic elements. This ranking-driven strategy is enhanced by chaotic dynamics, significantly improving diversity and preventing premature convergence. In the straight-line phase, when the transition angle... Beyond π, the geese switch to a straight formation. There is a 50% probability that they will either follow a strong peer with chaotic perturbations or perform a quantum rotation gate operation.

[0052] To further enhance development capabilities, the quantum rotating door operation is applied periodically every three iterations across the entire population, regardless of the current stage. This ensures that the solution is continuously refined throughout the optimization process.

[0053] To further demonstrate the practicality of CQ-SGA, this invention applies it as an optimizer to the PINN architecture search process. PINN has become a powerful method for solving partial differential equations by directly encoding the governing physical laws into the loss function of the neural network. While significant progress has been made, research on PINN variants highlights the crucial role of network architecture design in improving the accuracy, convergence speed, and robustness of complex problems. PINN architecture search is a discrete optimization problem, where the parameters of the neural network structure must be represented as discrete values. However, CQ-SGA was originally designed for continuous space optimization. To apply CQ-SGA to the PINN architecture search problem, a transformation mechanism is needed to discretize the continuous search space.

[0054] Continuous chaotic mappings are not suitable for discrete optimization. This invention uses discrete tent graphs for population initialization and local perturbation, the definition of which is given in Equation 8: The continuous rotation operation is redefined as a probabilistic bit flip, applied to the binary representation of the integer-encoded solution. The flip probability is defined in Equation 9: Where P flip It is the probability of flipping a single bit in binary encoding. These are dynamic control parameters.

[0055] This invention employs an integer encoding scheme, using formula 10 to map the continuous positions of the snow goose into discrete integers: Where x real These are coordinates in a continuous search space.

[0056] The fitness function of PINN is defined in Equation 11: Although the representation and core operators are discrete, the overall iterative process of CQ-SGA remains consistent with its continuous correspondence process. The algorithm retains its basic two-stage structure, fitness-based hierarchical updates, and adaptive switching mechanism. Pseudocode for the PINN search is given in Algorithm 2. Figure 2 The PINN architecture search workflow is described, comprising five key components. The initialization phase employs chaotic mapping and quantum encoding. The optimization loop executes snow goose flocking operations, including leader-follower updates, QRG fine-tuning, and chaotic local search. The search space defines the architecture parameters. The evaluation phase computes the PDE loss through forward propagation and fitness scoring. Termination checks and outputs the final architecture when the criteria are met. The experimental results and analysis are as follows: Benchmark Functions: This section uses the benchmark function sets from IEEE CEC 2017 and CEC 2022. CEC 2017 includes 30 functions: F1-F3 (unimodal with a single global optimum), F4-F10 (basic multimodal with multiple local optima), F11-F20 (hybrid combinations), and F21-F30 (nonlinear composite functions). CEC 2022 has 12 functions: F1 (unimodal), F2-F5 (multimodal), F6-F8 (hybrid), and F9-F12 (combination), with enhanced challenges such as asymmetric search space and variable linkage.

[0057] Comparison of algorithms: In order to rigorously evaluate the core optimization capabilities of the proposed algorithm CQ-SGA, comparative experiments were conducted against established metaheuristics. These included the original Snow Goose Algorithm (SGA), seven original algorithms (PSO [2], GA

[28] , DE

[29] , GWO

[30] , HHO

[31] , WOA

[32] , SA [4]) and five advanced algorithms (AEO

[33] , Augmented-AEO

[34] , MRFO

[35] , ES

[36] , SCA

[37] ).

[0058] Experimental setup: Evaluation was conducted on the CEC 2017 benchmark with a dimension of 30 and the CEC 2022 benchmark with a dimension of 10. Additional experiments were performed to validate competitiveness under high complexity. The CEC 2017 benchmark was re-evaluated at 50 dimensions, while the CEC 2022 benchmark was tested at 20 dimensions. The CQ-SGA parameters were fixed as follows: population size = 50, maximum iterations = 1500, and 3 independent runs. Parameter settings for all original and advanced algorithms are provided in Tables 1 and 2 for consistency.

[0059] Table 1. Basic Algorithm Parameter Settings Table 2. Advanced Algorithm Parameter Settings The performance comparison analysis of the algorithm of this invention specifically includes the following: To provide a comprehensive evaluation of CQ-SGA, this invention presents a performance comparison analysis of low-dimensional and high-dimensional benchmark suites. Specifically, Section 4.2.1 details the experimental results and comparative evaluation of the original and advanced algorithms on low-dimensional test suites (CEC 2017 30D and CEC 2022 10D). Subsequently, Section 4.2.2 extends this analysis to higher-dimensional problems (CEC 2017 50D and CEC 2022 20D) to examine the scalability and robustness of CQ-SGA under increased complexity. This structured approach helps to clearly understand the effectiveness of the algorithm at different problem scales.

[0060] Comparison with the original algorithm: Table 3 and Figure 3Experimental results and fitness convergence curves for several specific functions are presented. On the 30-dimensional CEC 2017 benchmark set, CQ-SGA was compared with seven original algorithms. For the unimodal function F1, CQ-SGA achieved an optimal solution of 7.74 × 10^3, significantly outperforming the suboptimal PSO (4.07 × 10^5), with an absolute difference of 3.99 × 10^5. For the multimodal function F5, all algorithms reached the theoretical optimum of 500, but CQ-SGA's standard deviation (5.32E-05) was 24.8% lower than GA's (7.07E-05), demonstrating superior stability. For the mixed function F18, CQ-SGA (1.38E+04) improved by 187% compared to PSO (3.96E+04). On the composite function F29, it outperforms PSO (2.60E+04) by 20.9%, with a value of 2.15E+04. Overall, the mean error of CQ-SGA (1.08E+04) is only 9.2% of that of the second-ranked PSO (1.18E+05), achieving the best ranking across 100% of the test functions. Table 4 and Figure 4 Experimental results and fitness convergence curves for several specific functions are presented. In the 10-dimensional CEC2022 test for the same original algorithm, PSO achieved optimality with a smaller standard deviation (4.02E-14) on the unimodal function F1, while CQ-SGA (3.00E+02) tied in function value. On the multimodal function F2, CQ-SGA (4.00E+02) outperformed PSO (4.04E+02) by 1%. Both achieved 2.03E+03 on the mixed function F7, but CQ-SGA's standard deviation (1.01E+01) was 21.1% higher than PSO's (1.28E+01). For the combined function F11, CQ-SGA's standard deviation (9.43E-03) was 81.2% lower than GA's (5.01E-02). Overall results show that the average error of CQ-SGA (1.33E+03) is comparable to that of PSO, but it achieves the best ranking in 75% of the test functions.

[0061] Table 3. AVG and STD (30D) of CQ-SGA and the original algorithm on specific functions in CEC 2017. Table 4. AVG and STD(10D) of CQ-SGA and the original algorithm on specific functions in CEC 2022. Comparison with advanced algorithms: Table 5 and Figure 5 Experimental results and fitness convergence curves for several specific functions are presented. CQ-SGA demonstrates its advantages over advanced algorithms.

[0062] On F12, its optimal solution (4.07E+03) outperforms the second-best Aug-AEO (1.78E+04) by 77.1%. For F23, it leads with 2.50E+03, exceeding the second-best ES (2.58E+03) by 3.1%. Although the averages of F2 / F5 match those of ES and MRFO, their standard deviations are 62.3% (F2) and 90.3% (F5) lower. Overall, the average score of CQ-SGA (2.73E+04) is 20.4% lower than the runner-up ES (3.43E+04), while ES's average score is 25.6% higher, firmly holding first place. Table 6 and... Figure 6 Experimental results and fitness convergence curves for several specific functions are presented. CQ-SGA maintains its advantage in the CEC 202210D advanced benchmark. It achieves the optimal solution for F6 (1.85E+04), exceeding the second-best MRFO (2.09E+04) by 11.5%. On F12, it leads with 2.75E+03, exceeding the second-best AEO (2.77E+03) by 0.7%. For F1 / F3, its mean matches Aug-AEO and ES, but with a lower standard deviation (e.g., F1: 2.71E-14 vs. Aug-AEO's 5.68E-14). Overall, CQ-SGA's mean (2.82E+03) is 12.7% lower than ES (3.23E+03), while ES's mean is 14.5% higher, giving it the overall top ranking. Table 5. AVG and STD (30D) of CQ-SGA and advanced algorithms on specific functions in CEC 2017 Table 6. AVG and STD(10D) of CQ-SGA and advanced algorithms on specific functions in CEC 2022. Comparison of the present invention with the original algorithm: Table 7 and Figure 7Experimental results and fitness convergence curves for several specific functions are presented. CQ-SGA demonstrates a significant advantage in the 50D problem. In F18, its optimal solution (7.44E+03) outperforms the second-best PSO (7.73E+03) by 3.8%. For F29, it leads by 1.69E+05, exceeding PSO (3.02E+05) by 44.0%. Although the average value (5.00E+02) in F1 matches that of PSO, its STD (5.84E-05) is 1.8% lower than the second-best GA (5.95E-05). Overall, the CQ-SGA's overall average value (3.66E+04) is 97.4% lower than the runner-up GA (1.41E+06), while the GA's average value is significantly higher. Compared to the 30D results (Table 3), the lead advantage increased from 90.8% to 97.4%, confirming enhanced efficacy at a higher dimension. (Table 8 and...) Figure 8 Experimental results and fitness convergence curves for several specific functions are presented. In the 20D test, CQ-SGA achieved the optimal solution for F2 (4.11 × 10^2), outperforming the second-best PSO (4.19 × 10^2) by 1.9%. For F7, it leads with 1.82E+03, exceeding PSO (1.99E+03) by 8.5%. Although F11 (2.60E+03) matches the average of GA, its STD (1.86E-02) is 56.7% better than GA (4.30E-02). The overall average (3.62E+03) is 37.9% lower than the runner-up GWO (5.83E+03), while GWO's average is 61.2% higher. Compared to 10D (Table 4), the lead increased from 0.262% to 37.9%, validating the enhanced robustness of medium-sized systems. Table 7. AVG and STD (50D) of CQ-SGA and the original algorithm on specific functions at CEC 2017. Table 8. AVG and STD of CQ-SGA and the original algorithm on CEC 2022 specific functions (20D) Comparison of this invention with advanced algorithms: Table 9 and Figure 9 Experimental results and fitness values ​​for several specific functions are presented respectively. Convergence Curves. CQ-SGA maintained its dominance over advanced algorithms in 50D. In F11, its optimal solution (1.74E+05) outperformed the second-best Aug-AEO (3.78E+05) by 54.0%. For F25, its mean (3.89E+03) matched ES, but its standard deviation was 30.7% higher (9.36E+01 vs. 1.35E+02). Overall, the mean (4.47E+04) was 53.3% lower than the August AEO (9.57E+04), while the August AEO mean was 114% higher. Compared to 30D (Table 5), the lead gap widened from 20.4% to 53.3%, demonstrating the enhanced effectiveness of chaotic quantum mechanisms in high-dimensional spaces. Table 10 and... Figure 10 Experimental results and fitness convergence curves for several specific functions are presented. In the 20D advanced benchmark, CQ-SGA achieved the best solution for F6 (2.63E+04), outperforming the second-best Aug-AEO (4.89E+04) by 46.2%. At F12, it leads with 2.98E+03, 0.7% higher than the August AEO (3.00E+03). For F1 / F3, its mean matches that of Aug-AEO, but shows a higher standard deviation (e.g., F1: 5.28E-08 vs. 5.81E-08). The overall mean (7.54E+03) is 42.9% lower than the August AEO mean (1.32E+04), which is 74.9% higher. Compared to 10D (Table 6), the lead advantage increased from 12.7% to 42.9%, demonstrating a stronger adaptability to high-dimensional problems.

[0063] Table 9. AVG and STD (50D) of CQ-SGA and advanced algorithms on specific functions in CEC 2017. Table 10. AVG and STD(20D) of CQ-SGA and advanced algorithms on specific functions in CEC 2022. The CQ-SGA algorithm demonstrated comprehensive advantages in the CEC 2017 / 2022 benchmark tests, significantly outperforming mainstream methods in solution quality and stability. From low-dimensional (10D / 30D) to high-dimensional (20D / 50D), its overall ranking remained consistently first, with even greater advantages at higher dimensions: leading the baseline algorithm by 97.4% in 50D (a 6.6 percentage point improvement over 30D) and outperforming state-of-the-art methods by 32.9%. The standard deviation of the chaotic quantum was 30% lower than the runner-up, effectively suppressing parametric oscillations. Notably, for complex 20D functions, it reduced the solution error by 46.2% compared to suboptimal methods, establishing a new performance benchmark for multidimensional optimization.

[0064] This invention quantifies the individual and combined contributions of the chaotic mapping (C) and quantum rotation gate (Q) components in the SGA. The evaluation covers two benchmark datasets: CEC 2017 with 30-dimensional functions and CEC2022 with 20-dimensional functions, measuring the average error of all functions in each benchmark. The study examines four algorithmic variants: the original SGA without enhancement, C-SGA (integrating C for population initialization), Q-SGA (integrating Q for solution updating), and CQ-SGA (a complete mixture of C and Q).

[0065] Figure 11 illustrates the clear performance hierarchy across the two benchmarks. The original SGA performs the worst, with an error of 6.364 × 10⁸ on CEC 2017 and 1.674 × 10⁸ on CEC 2022, demonstrating fundamental limitations in population diversity and convergence behavior. Q-SGA shows modest improvements on CEC 2017 at 3.751 × 10⁸ (a 41.1% reduction compared to SGA) and on CEC 2022 at 5.555 × 10⁷ (a 66.8% reduction), although quantum operations alone prove insufficient for efficient initialization at higher dimensions. The C-SGA results were significantly better, achieving 2.384 × 10^8 (a reduction of 62.5%) at CEC 2017 and 2.027 × 10^7 (a reduction of 87.9%) at CEC 2022, highlighting the key role of chaotic mapping in enhancing exploration capabilities. The fully mixed CQ-SGA offered superior performance, achieving 8.117 × 10^7 (an 87.2% reduction compared to SGA) at CEC 2017 and 6.416 × 10^6 (a 96.2% reduction) at CEC 2022. This translates to an improvement of 6.60 × 10^7 over C-SGA and 2.939 × 10^8 over Q-SGA at CEC 2017. The synergistic value of combining the two components was confirmed.

[0066] Figure 12 provides further insights by categorizing the results for different function types. Across all categories, multimodal, mixed-mode, single-mode, and composite functions, the performance hierarchy consistently remains SGA > Q-SGA > C-SGA > CQ-SGA. This pattern confirms the robustness of the findings of this invention, demonstrating that chaotic mappings offer significant benefits for exploring crucial multimodal and composite functions, while quantum rotation gates excel in single-modal functions requiring precise convergence. Most importantly, CQ-SGA maintains its advantage in every category, showcasing the adaptability of hybrid methods to diverse optimization challenges.

[0067] This ablation study ultimately demonstrates that both chaotic mapping and QRG significantly enhance SGA performance. The results highlight the crucial role of component interactions in hybrid algorithms, where synergistic composition consistently outperforms individual enhancements.

[0068] This invention employs sensitivity analysis to assess how variations in key parameters affect algorithm performance, identifying optimal configurations and robustness boundaries. The experiment examines two quantum-inspired parameters: the chaotic map iteration (Nchaos) controlling the complexity of the chaotic sequence during population initialization, and the rotation angle (θ) determining the magnitude of quantum state updates. Parameter configurations are systematically varied, with Nchaos tested at {1, 3, 5, 7, 9} and θ tested at {1, 2, 3, 4, 5}, using average loss as the performance metric. Figure 13 illustrates the results of the sensitivity analysis.

[0069] N chaos Sensitivity analysis revealed a distinct S-shaped performance curve. At Nchaos = 3, the algorithm achieved a minimum average loss of 0.0025, 46.8% lower than the 0.0047 loss at Nchaos = 1. This improvement stems from sufficient chaotic development, which enhances population diversity without compromising gradient information. However, increasing the number of iterations beyond this optimum leads to a significant performance drop: at Nchaos = 5, the loss jumps to 0.0090 (260% higher than the minimum), peaks at 0.0120 at Nchaos = 7 (a 380% decrease), and recovers slightly to 0.0095 at Nchaos = 9 (still 280% worse than the optimum). This pattern suggests that while insufficient chaos (Nchaos < 3) limits exploration, excessive iteration (Nchaos > 5) introduces destructive randomness, overriding the appropriate optimization signal. The narrow optimum window near Nchaos = 3 demonstrates the critical balance required between ergodicity and stability in chaotic initialization.

[0070] for The parameters and loss landscape exhibit complex, non-monotonic behavior. Performance drops sharply at the midpoint, with a maximum loss of 0.0120 at θ=2, compared to... The algorithm achieves a global minimum of 0.0050, which is 140% higher. It exhibits a local minimum of 0.0075, which is still 50% worse than the optimal configuration. It is worth noting that changing the step size from... (Loss = 0.0080 increased to) This achieves a 37.5% performance improvement, which contradicts classic expectations. This is due to the small step size (…). This leads to slow convergence and susceptibility to noise, while a medium step size ( This will lead to near-optimal oscillatory behavior. The superior performance of the time-space system utilizes quantum advantages; larger rotations can achieve more efficient energy barrier crossing through enhanced tunneling effects, resulting in lower losses compared to traditional methods. In the worst case, performance is 58.3% lower.

[0071] Based on comprehensive sensitivity analysis, this invention establishes the following optimal parameter selection: N chaos = 3 and = 5. The parameter configuration represents the optimal operating point for CQ-SGA.

[0072] The Burgers equations are a cornerstone of fluid mechanics, governing nonlinear phenomena ranging from shock waves to turbulence modeling

[38] . Their challenging combination of nonlinear advection and viscous diffusion makes high-fidelity solutions crucial for advancing our understanding of the physics

[39] . While high-precision numerical solutions exist, they often require complex spatial approximations and careful handling of time integrals to address sharp gradients, especially at high Reynolds numbers

[40] . Furthermore, obtaining solutions with multiple fronts through analysis remains a complex task, highlighting the inherent nonlinearity of the equations

[41] . To rigorously evaluate the ability of the CQ-SGA-optimized PINN framework to capture such complex physics, this invention deploys it on this typical problem. Experiments include a comparative analysis of CQ-SGA and five original algorithms (PSO, GA, GWO, DE, CS). The CQ-SGA was configured with the following parameters: population size = 50, 40 coarse search iterations, 1000 fine search iterations, and 3 independent runs. The parameter settings for all baseline algorithms are provided in Table 1. All algorithms were integrated into the DeepXDE-based PINN solver via function discretization.

[0073] Figure 14 The results of the coarse search phase are shown, where CQ-SGA achieves a significant loss of 6.528e-2, which is 35.5% better than the second-best algorithm GA (1.012e-1), demonstrating excellent initial optimization capabilities. It discovers an efficient three-layer PINN architecture (2→32→32→1) that strikes a balance between computational efficiency and expressive power. Figure 15 Key visual validation is provided: the CQ-SGA-PINN solution (b) achieves significant consistency with the ground reality (a), particularly in maintaining boundary layer integrity. In stark contrast, the PSO-optimized solution (c) exhibits non-physical oscillations near the boundary, indicating insufficient representational capability or optimization difficulties. This visually underscores the success of CQ-SGA in encoding physically consistent behavior. Figure 14The results of the fine-search phase are presented. After 1000 fine-search iterations, CQ-SGA achieved a final loss of 9.9873e-4 at epoch 1000, a 98.47% reduction compared to the coarse search result, validating the effectiveness of the two-stage optimization strategy. Comparative analysis reveals the superiority of CQ-SGA, as it outperforms all baselines by 40.7% (1.6859e-3) compared to GA, and by 93.6% compared to the weakest performer, CS (1.5687e-2). Crucially, CQ-SGA is the only algorithm to break the 1e-3 loss barrier, achieving sub-thousandth accuracy (9.9873e-4), while PSO and GWO remain at the 0.1% level. These results confirm CQ-SGA's dual advantages in architecture design and optimization stability in solving the Burgers equations, establishing a new state-of-the-art benchmark for PINN.

[0074] Tables 11 to 18 in the appendix of this invention provide a systematic comparison of the CQ-SGA algorithm on different benchmark sets. Specifically, Tables 11 and 12 provide results for the original algorithm and the advanced algorithm on the CEC 2017 30D problem, respectively. Tables 13 and 14 detail the performance on the CEC 2022 10D problem. Tables 15 and 16 show the results for the CEC 2017 50D problem, and Tables 17 and 18 show the results for the CEC 2022 20D problem. Each table includes the average performance metric (AVG) and its variability. Table 11: AVG and STD (30D) of CQ-SGA and the original algorithm on CEC 2017. Table 12 AVG and STD (30D) of CQ-SGA and advanced algorithms at CEC 2017 Table 13 AVG and STD(10D) of CQ-SGA and the original algorithm at CEC 2022 Table 14 AVG and STD(10D) of CQ-SGA and advanced algorithms at CEC 2022 Table 15 AVG and STD (50D) of CQ-SGA and the original algorithm at CEC 2017 Table 16 AVG and STD (50D) of CQ-SGA and advanced algorithms at CEC 2017 Table 17 AVG and STD (20D) of CQ-SGA and the original algorithm at CEC 2022 Table 18 AVG and STD (20D) of CQ-SGA and advanced algorithms at CEC 2022

Claims

1. A chaotic quantum enhanced snow goose algorithm for complex optimization problems, characterized by, The method includes the following steps: Step 1: Initialize the snow goose flock. Use the chaotic mapping method to randomly generate the initial position and velocity of each snow goose in the solution space to ensure that the distribution of the population in the search space has good ergonomics and diversity, and avoid the aggregation problem that may be caused by traditional random initialization. Step 2: Calculate the fitness value of each snow goose in the population. According to the type of optimization problem, compare the fitness of all snow geese, find the snow goose with the best fitness in the current population, and record its position as the global optimal solution. Step 3: Check if the algorithm has reached the preset termination condition. If it is met, the algorithm ends and outputs the global optimal solution; if it is not met, enter the main loop and continue to execute Step 4. Step 4: Based on the core formula of the Chaotic Quantum Enhanced Snow Goose Algorithm, and combining the snow goose's own historical best position with the population's global best position, update the flight speed and position of each snow goose. Step 5: This step introduces a multi-mode development mechanism based on snow goose flight formations. It determines whether a "V-formation" strategy is currently being adopted. If it is a V-formation, the V-formation development strategy is executed. In this mode, snow geese mainly utilize information around the global optimal solution for in-depth mining. At the same time, there may be a leader-follower relationship among snow geese similar to that of a V-formation goose formation, focusing on local development. If it is not a V-formation, the V-shaped development strategy is executed. This mode may simulate another group cooperation mode, balancing the exploration of individual optimal experience and group experience by snow geese, or focusing on exploration and development in different directions. Step 6: Perform boundary checks on the updated snow goose positions. If the snow goose position exceeds the preset solution space boundary, use a specific boundary processing strategy to correct it to the legal range, ensuring that the search is conducted within the feasible domain. Step 7: Regardless of which development mode was used in Step 5, a linear development strategy will be executed thereafter. This mode simulates the straight flight of geese in a flock, guiding the snow geese to search more directly and efficiently along the current optimal direction, further enhancing the convergence ability of the algorithm. Step 8: Recalculate the fitness values ​​of all snow geese after updating their positions in Steps 4-7, update the historical best position of each individual snow goose, and compare and update the global best position and corresponding best fitness value of the entire population. Step 9: Perform a quantum rotation operation once every 3 iterations. If the quantum rotation operation is not performed, proceed to step 10. This operation draws on the concept of quantum computing and updates the position of the snow goose by performing a specific "rotation". It may be intended to help the snow goose escape local optima, increase population diversity, and enhance global exploration capabilities. Step 10: Set the iteration counter t = t + 1. Then, the algorithm jumps back to step 3 and starts a new round of iterations until the termination condition is met.

2. The chaos quantum enhanced snow goose algorithm for complex optimization problems as claimed in claim 1 wherein, The method employs a discrete tent graph for population initialization and local perturbation, the definition of which is given in Equation 8: where b is the bit width, is the maximum integer value, clipped to the range , denotes the current integer encoding position value.

3. The chaos quantum enhanced snow goose algorithm for complex optimization problems as claimed in claim 1 wherein, The continuous rotation operation in the method is redefined as a probabilistic bit flip, applied to the binary representation of the integer-encoded solution, with the flip probability defined in Equation 9: where P flip is the probability of flipping a single bit in a binary encoding, is a dynamic control parameter.

4. The chaos quantum enhanced snow goose algorithm for complex optimization problems as claimed in claim 1 wherein, The method employs an integer encoding scheme, using formula 10 to map the continuous positions of the snow geese to discrete integers: where x real is a coordinate in the continuous search space.

5. The chaos quantum enhanced snow goose algorithm for complex optimization problems as claimed in claim 1, wherein, The fitness function of the method PINN is defined in Equation 11: wherein denotes the PDE residual loss, denotes the boundary condition loss, is a balancing weight.

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