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.

翻译：作为一类重要的相场方程，Cahn-Hilliard方程包含空间高阶导数、强非线性甚至奇异性。当使用物理信息神经网络（PINN）模拟长时间演化时，必须对时间域进行分解以捕捉不同时间段的解过渡。此外，基准PINN无法维持方程的质量守恒特性。我们提出了一种保持质量守恒的时空自适应PINN方法。该方法根据能量衰减速率自适应划分时间域，并在每个时间步使用独立神经网络求解Cahn-Hilliard方程。为提高预测精度，子域中采用空间自适应采样选取残差值较大的点，并将其加入训练样本。同时，在损失函数中引入质量约束以补偿PINN方法求解Cahn-Hilliard方程时的质量退化问题。通过一系列数值算例验证该保持质量守恒的时空自适应PINN方法的有效性，包括不同体势能的Cahn-Hilliard方程、含奇异性的三维Cahn-Hilliard方程以及Cahn-Hilliard方程组。数值结果证明了所提算法的有效性。