On the simulation of fractional Riccati equations with physics-informed neural networks

The Riccati equations are a classical type of nonlinear differential equations with important applications across mathematics, physics, and engineering. These equations are applicable in many domains, often linked to control theory, quantum mechanics, and stochastic processes. Due to the properties of fractional operators, the importance of Fractional Riccati differential equations may be observed, which are crucial in engineering disciplines, such as control theory, Resistor-Inductor electric circuits, and system dynamics, etc. In this study, we have developed the fractional model of the Riccati equations and proposed their solution. The fractional-order Riccati equation presents considerable difficulties owing to its inherent nonlinearity and nonlocal behavior. To overcome these challenges, this study utilizes the Physics-Informed Neural Networks approach, integrated with the predictor–corrector Method and optimized using the Adam algorithm. The proposed method is validated through two illustrative examples. The results are analyzed using both graphical and tabular representations. These graphical and numerical comparisons clearly demonstrate that the method offers superior accuracy. The resulting solution graphs reveal a strong agreement between the exact solutions and those obtained using Physics-Informed Neural Networks. It is observed that as the order of the fractional problem approaches an integer value, the solutions tend to converge toward those of the corresponding classical-order model.