A Memory-Enhanced Fractional Probabilistic Self-Organizing Map

Probabilistic Self-Organizing Maps (PRSOM) are effective for visualizing complex patterns in large datasets due to their neural network-based structure and probabilistic reasoning capabilities. However, the use of ordinary derivatives in their learning rules limits their ability to capture underlying temporal or spatial dependencies. To address this, we propose a novel model; Fractional Probabilistic Self-Organizing Map (FRAC-PRSOM), which integrates fractional derivatives into the PRSOM framework. Specifically, the Caputo–Fabrizio derivative of order [Formula: see text] is adopted to introduce memory and non-local behavior into the learning process. The model reformulates the learning rule to simultaneously incorporate both fractional calculus and probabilistic density estimation, thereby enhancing the system’s adaptability and depth of learning. We provide a theoretical analysis establishing the stability, sensitivity, and convergence of FRAC-PRSOM. The method exhibits near-linear scalability ([Formula: see text]). Comprehensive experiments on twelve benchmark datasets evaluate FRAC-PRSOM against seven state-of-the-art baselines using four clustering metrics. Results demonstrate statistically significant improvements ([Formula: see text]), with average Silhouette gains of 5–15% and up to 187% improvement on complex datasets (e.g., Magic). Statistical validation via Friedman and Nemenyi tests confirms FRAC-PRSOM’s superiority over all baselines in model fit while maintaining competitive cluster separation. Sensitivity analysis across [Formula: see text] reveals dataset-dependent optimal values while preserving numerical stability. Overall, FRAC-PRSOM provides a principled framework for memory-enhanced probabilistic clustering, offering particular promise for complex, high-dimensional datasets, where capturing non-local dependencies is crucial.