A neural network approach for composite fractional relaxation oscillation equations in complex dynamical systems

The relaxation oscillator is a system that operates on the tendency of physical processes to return to equilibrium after being disturbed. Its governing equation, the relaxation–oscillation equation, plays a central role in diverse fields such as physics, biology, and engineering. The nonlocal characteristic of fractional calculus operators make them ideal for systems with memory, hereditary effects, and long-range time dependence. Fractional models capture anomalous diffusion, relaxation, and oscillation more accurately than integer-order models. In this work, we have investigated a fractional-order model of the composite relaxation–oscillation equation, where the Riemann–Liouville fractional derivative is employed to accurately capture memory-dependent dynamics and anomalous relaxation behaviors that are characteristic of many complex systems. For the investigation, the physics-informed neural network (PINN) framework is used, which integrates governing physical laws with the flexibility of deep learning and effectively captures the dynamics with accuracy and stability, in contrast to conventional numerical methods that struggle with the nonlocality and complexity of fractional derivatives. Comparison of results with other approaches shows that the approach of PINN is more reliable and accurate compared to those produced by previously studied methods.