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

Area of Science:

• Computational Mathematics
• Applied Physics
• Machine Learning

Background:

• Fractional diffusion equations model complex phenomena but are challenging to solve.
• Inferring time-dependent fractional orders from data is crucial for accurate modeling.
• Existing methods often struggle with noisy or sparse datasets.

Purpose of the Study:

• To develop a novel method for simultaneously solving variable-order time fractional diffusion equations and inferring the fractional order.
• To leverage physics-informed neural networks (PINNs) for robust data assimilation.
• To demonstrate the method's effectiveness with various data conditions.

Main Methods:

• A dual-network architecture using PINNs was implemented.
• One network approximates the solution u(x,t).
• A second network learns the time-dependent fractional order, embedding governing equations into the loss function.

Main Results:

• Achieved high accuracy with mean squared errors below 10-4 for solutions and 10-3 for fractional orders in smooth cases.
• Demonstrated robustness in handling sparse, noisy, and non-smooth fractional order data.
• Validated the effectiveness of the PINN-based approach for inverse problems in fractional calculus.

Conclusions:

• The proposed PINN approach offers a flexible and accurate solution for variable-order time fractional diffusion equations.
• It effectively infers time-dependent fractional orders directly from data, even under challenging conditions.
• This method advances the application of machine learning in solving complex partial differential equations.