Novel intelligent exogenous neuro-architecture–driven machine learning approach for nonlinear fractional breast cancer risk system

Breast cancer remains one of the most prevalent and life-threatening diseases worldwide, necessitating mathematical modelling frameworks to capture the complexity of its progression and risk factors. This research endeavor uncovers the novel machine learning expedition using an Adaptive Nonlinear AutoRegressive eXogenous (ANARX) neural network on a Fractional Order Breast Cancer Risk (FO-BCR) model. A novel Caputo fractional operator-based breast cancer risk model is presented using a five compartmental system reflected by healthy, tumor, immune, estrogen, and fatty cells. A modified fractional Adams PECE method is opted to generate solutions of the five fractional order variants on the four diverse BCR scenarios. These temporal sequences are parsed as ground truth for the adept ANARX network, which is iteratively refined using the Levenberg-Marquardt (LM) algorithm. The performance evaluation of the temporal feature learning of the ANARX-LM algorithm is comprehensively evaluated against reference numerical outcomes using mean square error (MSE) performance graphics, input-error cross correlation, error autocorrelation, error histogram analysis, sequential response and comparative error analysis charts. Low disparity between reference solutions is observed for all FO-BCR systems, with MSE errors in the range of 10⁻⁸ to 10⁻¹¹. Finally, the ANARX-LM’s predictive prowess is evaluated using the single and multistep configurations. Minute errors in the range of 10⁻⁹ to 10⁻¹¹, 10⁻⁸ to 10⁻¹⁰ suggest accurate anticipation of the FO-BCR system enabling preventive and prognostic measures for breast cancer models. These empirical findings underscore the potential of advanced machine-learning-driven neuro-architecture for next-generation predictive-oncology solutions that may facilitate treatment strategies.