A new study explores the use of Physics-Informed Neural Networks (PINNs) enhanced with Monte Carlo methods to address fractional-order differential models. These models are fundamental for describing phenomena with memory effects, where the future state of a system depends not only on its current state but also on its past history. PINNs' ability to integrate physical laws directly into their machine learning architecture makes them a promising tool for solving complex differential equations, especially those lacking simple analytical solutions.

Fractional-order models find application in various areas of physics and engineering, including viscoelasticity, anomalous diffusion, and electrochemistry, where materials or systems exhibit non-local or memory-dependent behaviors. However, their numerical solution can be computationally intensive and challenging. The incorporation of Monte Carlo techniques into PINNs aims to improve the efficiency and accuracy in approximating solutions, particularly in scenarios where boundary conditions or system properties are uncertain or stochastic.

This hybrid approach leverages PINNs' ability to learn solution functions from data and physical constraints, while Monte Carlo contributes to a better exploration of the parameter space and uncertainty quantification. The development of these computational methodologies is crucial for advancing the understanding and design of complex systems with memory effects, opening new avenues for simulation and prediction in fields where classical integer-order models prove insufficient.