Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative.

Fractional diffusion equations with conformable derivative have become an important research topic in Newtonian mechanics, quantum mechanics, arbitrary time scale problems, diffusion transport, neutron dynamics and other fields. In this paper, we propose to study time fractional diffusion equations with conformable derivative by using physics-informed neural networks (PINNs) for the first time. By solving the supervised learning task, we design a new spatio-temporal function approximator with high data efficiency. L-BFGS algorithm is used to optimize our loss function, and back propagation algorithm is used to update our parameters to give the numerical solutions. Since the conformable derivative satisfies the Leibniz rule and the chain rule, we can directly derive the network through automatic differentiation without any additional numerical discretization or truncation. For the forward problem, we can take IC/BCs as the data, and use PINNs to solve the corresponding partial differential equation. Three numerical examples are carried out to demonstrate the effectiveness of our methods. In particular, when the order of the conformable fractional derivative α tends to 1, a class of weighted PINNs is introduced to overcome the accuracy degradation caused by the singularity of solutions. For the inverse problem, we use the obtained data to train the neural network and specify the estimation of the parameter λ in the equation. Numerical results show that the proposed method is easy to implement and can accurately identify parameters, even when the training data is corrupted by 1% uncorrelated noise. [ABSTRACT FROM AUTHOR]