Learning variable-order time fractional diffusion equations using Physics-Informed Neural Networks

Abstract

This paper introduces a novel approach using physics-informed neural networks (PINNs) to simultaneously solve variable-order time fractional diffusion equations and infer the time-dependent fractional order from data. By embedding the governing equations into the neural network’s loss function, our method achieves high accuracy and flexibility, even with sparse or noisy data. We present a dual-network architecture where one network approximates the solution u ( x , t ) while another learns the fractional order α ( t ) . Numerical experiments demonstrate the effectiveness of our approach, achieving mean squared errors below 10 −4 for solutions and 10 −3 for fractional orders in smooth cases, while also handling noisy data and non-smooth orders robustly.