Physics informed neural network based scheme and its error analysis for ψ-Caputo type fractional differential equations

This paper proposes a scientific machine learning approach based on Deep Physics Informed Neural Network (PINN) to solve ψ-Caputo-type differential equations. The trial solution is constructed based on the Theory of Functional Connection (TFC), and the loss function is built using the L1-based difference and quadrature rule. The learning is handled using the new hybrid average subtraction, standard deviation-based optimizer, and the nonlinear least squares approach. The training error is theoretically obtained, and the generalization error is derived in terms of training error. Numerical experiments are performed to validate the proposed approach. We also validate our scheme on the SIR model.