A Physics Informed Neural Network (PINN) framework for fractional order modeling of Alzheimer's disease

This study presents a novel fractional order model of Alzheimer's disease (mental disorder) using the Caputo derivative to accurately capture long term memory and hereditary effects in neurodegeneration. The mathematical model incorporates key pathological constituents including neurons, amyloid beta (<i>A</i> _β), tau proteins and microglial responses, allowing detailed simulation of their dynamic interactions. Fundamental properties of the model, including positivity, boundedness, invariant regions and equilibrium points, are rigorously analyzed to ensure biological feasibility. Sensitivity analysis identifies amyloid toxicity as the most influential driver of neuronal loss underscoring its central role in AD progression. Furthermore, a Physics Informed Neural Network (PINN) is developed to approximate system dynamics from noisy observations while ensuring compliance with biological and physical constraints. Compared to standard neural networks the PINN exhibits superior accuracy and robustness especially under data scarcity. By integrating fractional calculus, optimal control and machine learning, this work advances computational modeling of Alzheimer's disease and offers insights into therapeutic optimization.