To efficiently obtain optimal product design solutions that better align with user preferences by leveraging their distribution characteristics across design variables, this study proposes a novel variable preference surrogate model (VPSM) and an adaptive genetic strategy within an interactive differential evolution (IDE) framework. The core challenge in preference-driven optimization lies in accurately capturing subjective user intentions while minimizing the user’s evaluation effort, especially when design variables influence multiple objectives in complex ways. By explicitly differentiating how variables affect different objectives and integrating preference learning directly into the evolutionary mechanism, this study aims to bridge the gap between computational optimization and human-centered design.

The proposed approach begins by statistically analyzing the decision variables and classifying them into independent attributes, those affecting only a single objective, and correlated attributes, those influencing multiple objectives simultaneously. Gaussian functions are then used to model user preferences for each attribute type. Based on the feedback from user-evaluated solutions, preference degrees for independent and correlated attributes are inferred for unevaluated individuals, forming a VPSM to estimate fitness values and reduce computational or user evaluation costs. An adaptive genetic strategy dynamically adjusts crossover and mutation probabilities according to VPSM estimates and population state: crossover is decreased for high-fitness individuals to preserve quality solutions, whereas mutation is increased in low-diversity regions to promote exploration. The surrogate model is iteratively updated with new user feedback, continuously refining the preference model and guiding the evolutionary search toward solutions that better align with user intentions. As new user feedback is incorporated, the surrogate model is iteratively updated, continuously improving the model’s accuracy and guiding the evolutionary search toward solutions that better reflect user intentions over time.

To evaluate the performance of the proposed method, comparative experiments were conducted against two ablation algorithms, one without VPSM and another without the adaptive genetic strategy, as well as four state-of-the-art evolutionary optimization algorithms. These evaluations were performed on three widely used benchmark test functions and a real-world automotive side-profile design problem to ensure generalizability and practical relevance. The experimental results show that the proposed method consistently outperforms competing algorithms across multiple metrics, including convergence speed, solution quality, and stability. It reduces required user evaluations by up to 40% in some test cases, significantly alleviating user fatigue. Moreover, the proposed method demonstrates greater robustness and faster convergence in later search stages, confirming its effectiveness in refining solutions as additional preference information becomes available.

This study demonstrates that integrating the VPSM and adaptive genetic strategy within the IDE framework provides an efficient and effective solution for preference-driven product design optimization. By explicitly modeling user preferences across variable types and adapting the search accordingly, the proposed method reduces user fatigue while guiding successful optimization toward high-quality, user-aligned solutions. Validation on both synthetic benchmarks and a real-world engineering design scenario underscores its reliability. This study highlights the importance of combining variable-sensitive preference modeling with adaptive evolutionary operators to handle the complexity of human preferences in multi-objective settings. Future work will explore deep learning-based surrogate models and multi-user preference fusion to enhance the flexibility and scalability of the approach.

The signal of thin coatings overlaps in the time domain, making it challenging to directly apply the time-of-flight method for thickness detection. The accuracy of model-based methods combined with optimization algorithms depends on both the precision of the coating’s optical parameters and the modeling precision. These methods must also address the anisotropic properties of carbon fiber composite materials. However, in practice, the propagation path of terahertz signals changes only at the interface, and the sample response to terahertz signals can be regarded as approximating a linear system.

Since the front part of the terahertz signal waveform already contains critical interface information of the coating, and the response of the sample behaves as an approximate linear system, the propagation path of the terahertz signal changes only at the interface. Therefore, signal sparse decomposition combined with the time-of-flight method was employed to detect thin coating thickness on carbon fiber composite substrates. First, a comprehensive analysis of compressive sensing theory was conducted, identifying the spectral projection gradient algorithm with L1 norm as suitable, as it has been validated on isotropic bases. Experimental evidence clarified that the reflective terahertz time-domain systems are preferable for detecting coatings on substrates. Subsequently, based on practical coating thickness measurements, the principle of the SPGL1 algorithm was derived, and a perception matrix was constructed using reference signals. A coating thickness detection scheme combining signal sparse decomposition and time-of-flight method was proposed by analyzing the conditions required for solving convex optimal problems.

Experimental data demonstrated that when coating refractive indices differ significantly and the thickness is greater than or equal to 100 μm, the results, even with added noise, align well with ideal expectations. When coating refractive indices are similar but the thickness is greater than or equal to 100 μm, the SPGL1 method exhibits strong anti-interference capability.

This complete experimental design promotes a deeper understanding of fundamental theories and methods, such as time-domain spectroscopy, sparse decomposition, and the time-of-flight method. It bridges theoretical knowledge and practical application, fostering students’ ability to connect the two while cultivating their interest and skills in scientific research.

In sparse broad learning systems, the changing importance of output weights is overlooked. Some weights are unimportant in the early stages of model training but become important after being trimmed, making their recovery challenging. Inspired by dynamic sparse training in neural networks, this paper proposes a width learning system utilizing dynamic sparse training to compensate for pruning errors during model training and improve overall model performance while maintaining model sparsity.

This system introduces a regularization term to constrain the output weight threshold in the objective function of a standard-width learning system. It seeks optimal network parameters and a sparse network structure through joint training of output weights and their thresholds. Introduce an output weight threshold for each output weight, and generate an output weight mask for the control model structure based on changes in the importance of the output weight. The mask is jointly generated using weights and their thresholds and dynamically adjusts weight threshold during training to prune and restore output weights based on changes in weight importance. This system can indirectly sparsify models using the mask while retaining output weights, achieving an optimal balance between network structure and accuracy through dynamic training, and improving the overall model performance by minimizing the incorrect pruning of weights. There is the greatest improvement in accuracy on the dataset 'BUTCSP', with an increase of approximately 30.12%. This article introduces exponential powers as regularization terms to constrain the weight threshold in the loss function of a standard-width learning system and adds a weight mask to the error term of the loss function. The alternating direction multiplier method is used to optimize and solve the objective function.

To verify the effectiveness of the broad learning system based on dynamic sparse training (BLSDST), this paper uses six UCI public datasets for simulation. The performance of the system was compared with those of the broad learning system(BLS) and lasso broad learning system(L1BLS). Results indicate that the BLSDST achieves a balance between model accuracy and sparsity by constraining the weight-threshold regularization term. Further, it can reduce model complexity without sacrificing model accuracy while compensating for the impact of model pruning on model performance.

Experimental results show that the proposed system can achieve model dynamic sparsity without reducing model performance and even improving it.