An artificial neural network approach to identify the parameter in a nonlinear subdiffusion model

Abstract

In this paper, we propose an artificial neural network approach to identify the parameter in a non-linear subdiffusion model from additional data. Instead of determining the parameter in the time fractional diffusion model by its form itself, we approximate it in the form of an artificial neural network. The key point of this approach relies on the approximation capability of neural networks. We formulate this inverse problem as an optimal control one, and we demonstrate the existence of the solution for the control problem and provide a mathematical analysis and the derivation of optimal conditions. Moreover, various numerical tests of the regular and singular examples have shown that the artificial neural network method (ANN) is effective. This is reinforced by its numerical comparison with the gradient descent, the alternating direction multiplier method (ADMM), physics-informed neural network (PINN) and DeepONet method.