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

TL;DR

This paper introduces a physics-informed neural network approach to solve forward and inverse conformable time fractional diffusion equations, demonstrating high accuracy and robustness even with noisy data.

Contribution

It is the first to apply PINNs to conformable fractional diffusion equations, proposing a new spatio-temporal approximator and weighted PINNs for improved accuracy.

Findings

Effective solutions for forward problems with IC/BCs

Accurate parameter estimation in inverse problems with noisy data

Weighted PINNs mitigate accuracy loss as derivative order approaches 1

Abstract

Physics-informed neural networks (PINNs) show great advantages in solving partial differential equations. In this paper, we for the first time propose to study conformable time fractional diffusion equations by using PINNs. By solving the supervise 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 our numerical solutions. For the forward problem, we can take IC/BCs as the data, and use PINN to solve the corresponding partial differential equation. Three numerical examples are are carried out to demonstrate the effectiveness of our methods. In particular, when the order of the conformable fractional derivative $\alpha$ 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 data obtained to train the neural network, and the estimation of parameter $\lambda$ in the equation is elaborated. Similarly, we give three numerical examples to show that our method can accurately identify the parameters, even if the training data is corrupted with 1\% uncorrelated noise.