A novel numerical approach for time-varying impulsive fractional differential equations using theory of functional connections and neural network

In this paper, we propose a physics-informed neural network-based scheme to solve time-varying impulsive fractional differential equations without any labeled data. At first, the existence and uniqueness of the solution of the impulsive fractional system are proved theoretically using the Bourdin state transition matrix-based solution representation and the Banach fixed point theorem. The proposed method uses the physics-informed neural network, extreme learning machines, and the theory of functional connections. A neural network is employed as the free function to construct the constrained expression. Based on the constrained expression, the linear system is trained using the extreme learning machine algorithm, and the nonlinear system is trained using the iterative least squares method, as the only trainable parameters are the output weights. The convergence and error analysis for the algorithm is derived. The neural network training is graphically demonstrated for the nonlinear system using the loss obtained in each iteration. For the linear case, the results are validated with the original solution, while for the nonlinear case, it is verified with the original solution and other classical methods. The primary use of the theory of functional connections is it helps the solution satisfy the physical constraint. •Existence of solution of time-varying IFDE is proved by Banach theory and STM.•PINN-XTFC approach for time-varying IFDE is developed.•The convergence and error analysis are derived in detail.•Error for certain examples is compared with the classical PINN-based solution.