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The physics-informed neural networks (PINNs) has been widely utilized to numerically approximate PDE problems. While PINNs has achieved good results in producing solutions for many partial differential equations, studies have shown that it does not perform well on phase field models. In this paper, we partially address this issue by introducing a modified physics-informed neural networks. In particular, they are used to numerically approximate Allen- Cahn-Ohta-Kawasaki (ACOK) equation with a volume constraint. Technically, the inverse of Laplacian in the ACOK model presents many challenges to the baseline PINNs. To take the zero- mean condition of the inverse of Laplacian, as well as the volume constraint, into consideration, we also introduce a separate neural network, which takes the second set of sampling points in the approximation process. Numerical results are shown to demonstrate the effectiveness of the modified PINNs. An additional benefit of this research is that the modified PINNs can also be applied to learn more general nonlocal phase-field models, even with an unknown nonlocal kernel. 

In this letter, we revisit the invariant energy quadratization (IEQ) method and provide a new perspective on its ability to preserve the original energy dissipation laws. The IEQ method has been widely used to design energy stable numerical schemes for phase-field or gradient flow models. Although there are many merits of the IEQ method, one major disadvantage is that the IEQ method usually respects a modified energy law, where the modified energy is expressed in the auxiliary variables. Still, the dissipation laws in terms of the original energy are not guaranteed by the IEQ method. Using the widely-used Cahn-Hilliard equation as an example, we demonstrate that the Runge-Kutta IEQ method indeed can preserve the original energy dissipation laws for certain situations up to arbitrary high-order accuracy. Interested readers are encouraged to extend this idea to more general cases and apply it to other thermodynamically consistent models.