The diversification of retrieved final neuron states through non-systematic satisfiability logical representation is pivotal to ensuring the optimality and functionality of Discrete Hopfield Neural Networks (DHNN) under varying neuron complexities. However, DHNN learning frameworks are predominantly designed for single-objective optimization, which often leads to repetitive neuron state patterns, overfitting, and limited storage capacity, particularly under increasing neuron complexity. These limitations indicate the need for a learning mechanism that can simultaneously enhance solution optimality and diversity. Motivated by this gap, we proposed a multi-objective DHNN framework based on a non-systematic Major 2 Satisfiability (MAJ2SAT) logical representation integrated with a Hybrid Exhaustive Search (HES) learning algorithm enhanced by an intelligent mutation operator. The proposed framework jointly optimized neuron fitness and neuron state diversity, enabling the systematic generation of high-quality and diversified neuron states. Unlike conventional exhaustive or heuristic search methods, the intelligent mutation operator selectively modified neuron states associated with unsatisfied clauses, thereby improving exploration efficiency while preserving solution feasibility. To further enhance the learning capability of DHNN, the proposed model introduced the concept of power strings, which facilitated the construction of multiple content addressable memories and effectively expanded the storage capacity of the network. Extensive experiments conducted on simulated datasets demonstrated that the proposed approach consistently I outperforms several state-of-the-art learning algorithms across problem sizes. The results showed reduced learning error in neuron diversity, increased total neuron variation, and a higher global minimum attainment ratio under varying clause configurations. Overall, the proposed multi-objective DHNN–MAJ2HES framework establishes a robust and scalable learning paradigm that enhances solution quality and diversity, with strong potential for extension to other logic-based neural optimization problems.
Citation: Alyaa Alway, Mohd Shareduwan Mohd Kasihmuddin, Mohd. Asyraf Mansor, Nur Ezlin Zamri, Yueling Guo, Siti Zulaikha Mohd Jamaludin, Azleena Mohd Kassim. Multi-objective optimization via hybrid exhaustive search with intelligent mutation for major 2 satisfiability in Discrete Hopfield Neural Networks[J]. AIMS Mathematics, 2026, 11(5): 15074-15119. doi: 10.3934/math.2026621
The diversification of retrieved final neuron states through non-systematic satisfiability logical representation is pivotal to ensuring the optimality and functionality of Discrete Hopfield Neural Networks (DHNN) under varying neuron complexities. However, DHNN learning frameworks are predominantly designed for single-objective optimization, which often leads to repetitive neuron state patterns, overfitting, and limited storage capacity, particularly under increasing neuron complexity. These limitations indicate the need for a learning mechanism that can simultaneously enhance solution optimality and diversity. Motivated by this gap, we proposed a multi-objective DHNN framework based on a non-systematic Major 2 Satisfiability (MAJ2SAT) logical representation integrated with a Hybrid Exhaustive Search (HES) learning algorithm enhanced by an intelligent mutation operator. The proposed framework jointly optimized neuron fitness and neuron state diversity, enabling the systematic generation of high-quality and diversified neuron states. Unlike conventional exhaustive or heuristic search methods, the intelligent mutation operator selectively modified neuron states associated with unsatisfied clauses, thereby improving exploration efficiency while preserving solution feasibility. To further enhance the learning capability of DHNN, the proposed model introduced the concept of power strings, which facilitated the construction of multiple content addressable memories and effectively expanded the storage capacity of the network. Extensive experiments conducted on simulated datasets demonstrated that the proposed approach consistently I outperforms several state-of-the-art learning algorithms across problem sizes. The results showed reduced learning error in neuron diversity, increased total neuron variation, and a higher global minimum attainment ratio under varying clause configurations. Overall, the proposed multi-objective DHNN–MAJ2HES framework establishes a robust and scalable learning paradigm that enhances solution quality and diversity, with strong potential for extension to other logic-based neural optimization problems.
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Figures(11) / Tables(9)
Alyaa Alway, Mohd Shareduwan Mohd Kasihmuddin, Mohd. Asyraf Mansor, Nur Ezlin Zamri, Yueling Guo, Siti Zulaikha Mohd Jamaludin, Azleena Mohd Kassim. Multi-objective optimization via hybrid exhaustive search with intelligent mutation for major 2 satisfiability in Discrete Hopfield Neural Networks[J]. AIMS Mathematics, 2026, 11(5): 15074-15119. doi: 10.3934/math.2026621
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