Convergence Properties of PINNs for the Navier-Stokes-Cahn-Hilliard System

Approximating solutions to differential equations using neural networks has become increasingly popular and shows significant promise. In this paper, we propose a simplified framework for analyzing the potential of neural networks to simulate differential equations based on the properties of the equations themselves. We apply this framework to the Cahn-Hilliard and Navier-Stokes-Cahn-Hilliard systems, presenting both theoretical analysis and practical implementations. We then conduct numerical experiments on toy problems to validate the framework's efficacy in accurately capturing the desired properties of these systems and numerically estimate relevant convergence properties.