Physics-informed neural networks for phase-field method in two-phase flow

The complex flow modeling based on machine learning is becoming a promising way to describe multiphase fluid systems. This work demonstrates how a physics-informed neural network promotes the combination of traditional governing equations and advanced interface evolution equations without intricate algorithms. We develop physics-informed neural networks for the phase-field method (PF-PINNs) in two-dimensional immiscible incompressible two-phase flow. The Cahn–Hillard equation and Navier–Stokes equations are encoded directly into the residuals of a fully connected neural network. Compared with the traditional interface-capturing method, the phase-field model has a firm physical basis because it is based on the Ginzburg–Landau theory and conserves mass and energy. It also performs well in two-phase flow at the large density ratio. However, the high-order differential nonlinear term of the Cahn–Hilliard equation poses a great challenge for obtaining numerical solutions. Thus, in this work, we adopt neural networks to tackle the challenge by solving high-order derivate terms and capture the interface adaptively. To enhance the accuracy and efficiency of PF-PINNs, we use the time-marching strategy and the forced constraint of the density and viscosity. The PF-PINNs are tested by two cases for presenting the interface-capturing ability of PINNs and evaluating the accuracy of PF-PINNs at the large density ratio (up to 1000). The shape of the interface in both cases coincides well with the reference results, and the dynamic behavior of the second case is precisely captured. We also quantify the variations in the center of mass and increasing velocity over time for validation purposes. The results show that PF-PINNs exploit the automatic differentiation without sacrificing the high accuracy of the phase-field method.