AFP-PINN: adaptive fractional polynomial-based physics-informed neural network for solving distributed-order time-fractional diffusion equations and its error analysis

Abstract This paper introduces a physics-based neural network framework for solving distributed-order time-fractional diffusion and diffusion-wave equations. The proposed framework utilizes physics-informed neural networks as trial solutions for these equations. To enhance the expressiveness of the network, we employ fractional Chebyshev orthogonal polynomials and introduce a new family of adaptive activation functions for problems with inherent fractional behavior. The integrals in the distributed-order terms are approximated using the Legendre-Gauss quadrature rule. The fractional derivatives are discretized by employing operational matrices. Initial and boundary conditions are enforced as both soft and hard constraints within the loss functions. The optimization process is carried out using the Adam optimizer. Furthermore, an error estimate is provided for the generalization error in terms of the quadrature rule approximation, neural network architecture, training errors, and the number of training samples. The proposed framework leverages the advantages of randomized sampling strategies, such as residual-adaptive refinement strategy. Numerical experiments demonstrate the efficiency and accuracy of our approach in terms of approximation errors and computational times, outperforming existing techniques.