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

TL;DR

This paper analyzes the convergence properties of Physics-Informed Neural Networks (PINNs) when applied to the Navier-Stokes-Cahn-Hilliard system, combining theoretical insights with numerical experiments to evaluate their effectiveness.

Contribution

It introduces a simplified analytical framework for PINNs applied to complex differential systems and validates it through theoretical analysis and numerical experiments.

Findings

PINNs can effectively approximate Navier-Stokes-Cahn-Hilliard solutions

The framework provides insights into convergence behavior of PINNs

Numerical experiments confirm theoretical predictions

Abstract

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.