Mass-preserving spatio-temporal adaptive PINN for Cahn-Hilliard equations with strong nonlinearities and singularities.

As one kind of important phase field equations, Cahn-Hilliard equations that involve high-order spatial derivatives, strong nonlinearities, and even solution singularities when certain bulk potentials (e.g., Flory-Huggins) are used. When using the physics informed neural network (PINN) to simulate the long time evolution of the solution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the standard PINN can't maintain the mass conservation property for the equations exactly. We propose a novel mass-preserving spatio-temporal adaptive PINN, which adaptively divides the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation within each subinterval. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value which are added to the training samples. Notably, a mass constraint is added to the loss function to compensate for the mass degradation problem of the PINN method when solving Cahn-Hilliard equations. Numerical experiments are presented to illustrate the effectiveness of the proposed method in solving complex phase fields models, including the Cahn-Hilliard equations with different bulk potentials, the three-dimensional Cahn-Hilliard equation with singularities, and the system of Cahn-Hilliard equations.