Adaptive loss weighting auxiliary output fPINNs for solving fractional partial integro-differential equations

We propose an adaptive loss weighting auxiliary output fractional physics-informed neural networks (AWAO-fPINNs) based on the fractional physics-informed network (fPINNs) for solving fractional partial integro-differential equations. In this framework, the automatic differentiation technique and numerical differentiation algorithm are effectively combined to construct a universal numerical scheme for Caputo fractional derivatives of different orders. Secondly, using the multi-output form of neural networks, the main output represents the required numerical solution and the auxiliary outputs represent the integrals in the equation. The relationships and new constraints between all outputs are obtained through automatic differentiation, successfully avoiding the discretization of the integrals. In addition, an adaptive loss weighting strategy is introduced into the model, which is based on the maximum likelihood estimation to continuously update the adaptive weights to realize the automatic allocation of loss weights, thus improving the accuracy of the model. Finally, we verify the effectiveness and accuracy of the AWAO-fPINNs model via several numerical experiments.