In this paper, the two-dimensional (2-D) fractional cable equation (FCE) with the Caputo variable-order (V-O) derivative was utilized for simulating systems with memory and hereditary characteristics that vary across time and space. This variable-order fractional model is particularly well suited for the description of neuronal dynamics in biological systems. The accurate modeling of dynamic, memory-dependent behaviors that vary over space and time, which are essential for applications such as neuronal dynamics, presents a challenge for conventional numerical methods. Furthermore, there is a lack of stable and effective numerical techniques for 2-D V-O systems, highlighting the need for improved computational approaches. In order to solve the cable equation numerically with high accuracy and computing efficiency, this work primarily focused on using a higher-order finite difference method. The proposed method's robustness was confirmed by stability and convergence analyses, while its efficacy was demonstrated by numerical simulations, which were presented in tabular and graphical formats. These findings demonstrate its precision and efficiency when dealing with the intricate dynamics of V-O fractional equations. The study concludes that the higher-order finite difference method offers an accurate and effective framework for solving fractional partial differential equations (FPDEs), particularly in applications that necessitate precision modeling, such as biological and physical systems. It also creates opportunities for future research, such as the application of the method to multivariate problems, the integration of machine learning techniques, or the adaptation of the method to systems with variable coefficients.

The current development of logic satisfiability in discrete Hopfield neural networks (DHNN)has been segregated into systematic logic and non-systematic logic. Most of the research tends to improve non-systematic logical rules to various extents, such as introducing the ratio of a negative literal and a flexible hybrid logical structure that combines systematic and non-systematic structures. However, the existing non-systematic logical rule exhibited a drawback concerning the impact of negative literal within the logical structure. Therefore, this paper presented a novel class of non-systematic logic called conditional random k satisfiability for k = 1, 2 while intentionally disregarding both positive literals in second-order clauses. The proposed logic was embedded into the discrete Hopfield neural network with the ultimate goal of minimizing the cost function. Moreover, a novel non-monotonic Smish activation function has been introduced with the aim of enhancing the quality of the final neuronal state. The performance of the proposed logic with new activation function was compared with other state of the art logical rules in conjunction with five different types of activation functions. Based on the findings, the proposed logic has obtained a lower learning error, with the highest total neuron variation TV = 857 and lowest average of Jaccard index, JSI = 0.5802. On top of that, the Smish activation function highlights its capability in the DHNN based on the result ratio of improvement Zm and TV. The ratio of improvement for Smish is consistently the highest throughout all the types of activation function, showing that Smish outperforms other types of activation functions in terms of Zm and TV. This new development of logical rule with the non-monotonic Smish activation function presents an alternative strategy to the logic mining technique. This finding will be of particular interest especially to the research areas of artificial neural network, logic satisfiability in DHNN and activation function.

Since the beginning of the Covid-19 infections in December 2019, the virus has emerged as the most lethally contagious in world history. In this study, the Hopfield neural network and logic mining technique merged to extract data from a model to provide insight into the link between factors influencing the Covid-19 datasets. The suggested technique uses a 3-satisfiability-based reverse analysis (3SATRA) and a hybridized Hopfield neural network to identify the relationships relating to the variables in a set of Covid-19 data. The list of data is to identify the relationships between the key characteristics that lead to a more prolonged time of death of the patients. The learning phase of the hybridized 3-satisfiability (3SAT) Hopfield neural network and the reverse analysis (RA) method has been optimized using a new method of fuzzy logic and two metaheuristic algorithms: Genetic and harmony search algorithms. The performance assessment metrics, such as energy analysis, error analysis, computational time, and accuracy, were computed at the end of the algorithms. The multiple performance metrics demonstrated that the 3SATRA with the fuzzy logic metaheuristic algorithm model outperforms other logic mining models. Furthermore, the experimental findings have demonstrated that the best-induced logic identifies important variables to detect critical patients that need more attention. In conclusion, the results validate the efficiency of the suggested approach, which occurs from the fact that the new version has a positive effect.

The discrete Hopfield neural network 3-satisfiability (DHNN-3SAT) model represents an innovative application of deep learning techniques to the Boolean SAT problem. Existing research indicated that the DHNN-3SAT model demonstrated significant advantages in handling 3SAT problem instances of varying scales and complexities. Compared to traditional heuristic algorithms, this model converged to local minima more rapidly and exhibited enhanced exploration capabilities within the global search space. However, the model faced several challenges and limitations. As constraints in SAT problems dynamically increased, decreased, or changed, and as problem scales expanded, the model's computational complexity and storage requirements may increase dramatically, leading to reduced performance in handling large-scale SAT problems. To address these challenges, this paper first introduced a method for designing network synaptic weights based on fundamental logical clauses. This method effectively utilized the synaptic weight information from the original SAT problem within the DHNN network, thereby significantly reducing redundant computations. Concrete examples illustrated the design process of network synaptic weights when constraints were added, removed, or updated, offering new approaches for managing the evolving constraints in SAT problems. Subsequently, the paper presented a DHNN-3SAT model optimized by genetic algorithms combined with K-modes clustering. This model employed genetic algorithm-optimized K-modes clustering to effectively cluster the initial space, significantly reducing the search space. This approach minimized the likelihood of redundant searches and reduced the risk of getting trapped in local minima, thus improving search efficiency. Experimental tests on benchmark datasets showed that the proposed model outperformed traditional DHNN-3SAT models, DHNN-3SAT models combined with genetic algorithms, and DHNN-3SAT models combined with imperialist competitive algorithms across four evaluation metrics. This study not only broadened the application of DHNN in solving 3SAT problems but also provided valuable insights and guidance for future research.

Within the swiftly evolving domain of neural networks, the discrete Hopfield-SAT model, endowed with logical rules and the ability to achieve global minima of SAT problems, has emerged as a novel prototype for SAT solvers, capturing significant scientific interest. However, this model shows substantial sensitivity to network size and logical complexity. As the number of neurons and logical complexity increase, the solution space rapidly contracts, leading to a marked decline in the model's problem-solving performance. This paper introduces a novel discrete Hopfield-SAT model, enhanced by Crow search-guided fuzzy clustering hybrid optimization, effectively addressing this challenge and significantly boosting solving speed. The proposed model unveils a significant insight: its uniquely designed cost function for initial assignments introduces a quantification mechanism that measures the degree of inconsistency within its logical rules. Utilizing this for clustering, the model utilizes a Crow search-guided fuzzy clustering hybrid optimization to filter potential solutions from initial assignments, substantially narrowing the search space and enhancing retrieval efficiency. Experiments were conducted with both simulated and real datasets for 2SAT problems. The results indicate that the proposed model significantly surpasses traditional discrete Hopfield-SAT models and those enhanced by genetic-guided fuzzy clustering optimization across key performance metrics: Global minima ratio, Hamming distance, CPU time, retrieval rate of stable state, and retrieval rate of global minima, particularly showing statistically significant improvements in solving speed. These advantages play a pivotal role in advancing the discrete Hopfield-SAT model towards becoming an exemplary SAT solver. Additionally, the model features exceptional parallel computing capabilities and possesses the potential to integrate with other logical rules. In the future, this optimized model holds promise as an effective tool for solving more complex SAT problems.

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.