The training phase of the random 3-satisfiability problem in discrete Hopfield neural networks aims to identify more satisfying clauses, enhancing synaptic weight for energy function computation and minimizing network energy. The primary challenge lies in designing synaptic weights that can be dynamically optimized to adapt to various formula structures while ensuring complete clause satisfaction and avoiding local optima. This enables an optimal balance between clause satisfaction and network convergence performance. To address this challenge, this paper first proposes a method for determining synaptic weights during the training phase based on the logical relationships between clauses and variables, simplifying the computation process and improving solution efficiency. Second, the hybrid firefly algorithm is employed during the training phase to optimize the number of satisfied clauses. This is achieved through a balance of global and local search mechanisms and a diversity maintenance strategy, facilitating the identification of more satisfied clauses, thereby leading to the generation of high-quality synaptic weights. Consequently, during the retrieval phase of the network, local field updates are executed based on these synaptic weights to find the optimal neuron states, thereby minimizing the energy function and improving global convergence performance. To evaluate the effectiveness of the hybrid firefly algorithm and the simplification of synaptic weight computation, we employed a comprehensive performance evaluation framework composed of maximum fitness, fitness ratio, entropy-adjusted diversity metrics, weight error, global minima ratio, energy error, similarity indices, and runtime. Experimental results indicate that the proposed model outperforms both traditional discrete Hopfield neural network random 3-satisfiability models and those that combine election algorithms with discrete Hopfield neural network random 3-satisfiability models across multiple performance metrics.
Citation: Xiaoya Chen, Saratha Sathasivam. A hybrid firefly algorithm for synaptic weight optimization in discrete Hopfield neural networks applied to random 3-satisfiability problems[J]. AIMS Mathematics, 2025, 10(6): 14840-14892. doi: 10.3934/math.2025667
The training phase of the random 3-satisfiability problem in discrete Hopfield neural networks aims to identify more satisfying clauses, enhancing synaptic weight for energy function computation and minimizing network energy. The primary challenge lies in designing synaptic weights that can be dynamically optimized to adapt to various formula structures while ensuring complete clause satisfaction and avoiding local optima. This enables an optimal balance between clause satisfaction and network convergence performance. To address this challenge, this paper first proposes a method for determining synaptic weights during the training phase based on the logical relationships between clauses and variables, simplifying the computation process and improving solution efficiency. Second, the hybrid firefly algorithm is employed during the training phase to optimize the number of satisfied clauses. This is achieved through a balance of global and local search mechanisms and a diversity maintenance strategy, facilitating the identification of more satisfied clauses, thereby leading to the generation of high-quality synaptic weights. Consequently, during the retrieval phase of the network, local field updates are executed based on these synaptic weights to find the optimal neuron states, thereby minimizing the energy function and improving global convergence performance. To evaluate the effectiveness of the hybrid firefly algorithm and the simplification of synaptic weight computation, we employed a comprehensive performance evaluation framework composed of maximum fitness, fitness ratio, entropy-adjusted diversity metrics, weight error, global minima ratio, energy error, similarity indices, and runtime. Experimental results indicate that the proposed model outperforms both traditional discrete Hopfield neural network random 3-satisfiability models and those that combine election algorithms with discrete Hopfield neural network random 3-satisfiability models across multiple performance metrics.
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Xiaoya Chen, Saratha Sathasivam. A hybrid firefly algorithm for synaptic weight optimization in discrete Hopfield neural networks applied to random 3-satisfiability problems[J]. AIMS Mathematics, 2025, 10(6): 14840-14892. doi: 10.3934/math.2025667
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