  As one kind important phase field equations, Cahn-Hilliard equations contain
spatial high order derivatives, strong nonlinearities, and even singularities.
When using the physics informed neural network (PINN) to simulate the long time
evolution, it is necessary to decompose the time domain to capture the
transition of solutions in different time. Moreover, the baseline PINN can't
maintain the mass conservation property for the equations. We propose a
mass-preserving spatio-temporal adaptive PINN. This method adaptively dividing
the time domain according to the rate of energy decrease, and solves the
Cahn-Hilliard equation in each time step using an independent neural network.
To improve the prediction accuracy, spatial adaptive sampling is employed in
the subdomain to select points with large residual value and add them to the
training samples. Additionally, a mass constraint is added to the loss function
to compensate the mass degradation problem of the PINN method in solving the
Cahn-Hilliard equations. The mass-preserving spatio-temporal adaptive PINN is
employed to solve a series of numerical examples. These include the
Cahn-Hilliard equations with different bulk potentials, the three dimensional
Cahn-Hilliard equation with singularities, and the set of Cahn-Hilliard
equations. The numerical results demonstrate the effectiveness of the proposed
algorithm.
