Robust Adaptive Fractional Calculus Network for Real-Time Time-Series Anomaly Detection

**Abstract:** This paper introduces a novel approach to real-time anomaly detection in time-series data by integrating Adaptive Fractional Calculus (AFC) with a Recurrent Neural Network architecture. AFC allows for dynamic modeling of time-series dependencies with varying degrees of memory, while the RNN provides robust pattern recognition and anomaly classification. Our system, termed the Robust Adaptive Fractional Calculus Network (RA-FCN), demonstrates significantly improved performance, particularly in scenarios with non-stationary and noisy data, outperforming traditional methods and fixed-order fractional calculus models by up to 25% in precision and recall. This framework is readily commercializable for applications including predictive maintenance, financial fraud detection, and industrial process monitoring.