Physics-informed neural networks for solving two-phase flow problems with moving interfaces

Abstract

In this paper, a meshfree method using physics-informed neural networks (PINNs) is developed for solving two-phase flow problems with moving interfaces, where two immiscible fluids bearing different material properties, are separated by a dynamically evolving interface and interact with each other through interface conditions. Two kinds of distinct scenarios of interface motion are addressed: the prescribed interface motion whose moving velocity is explicitly given, and the solution-driven interface motion whose evolution is determined by the velocity field of two-phase flow. Based upon piecewise deep neural networks and spatiotemporal sampling points/training set in each fluid subdomain, the proposed PINNs framework reformulates the two-phase flow moving interface problem as a least-squares (LS) minimization problem, which involves all residuals of governing equations, interface conditions, boundary conditions and initial conditions. Furthermore, approximation properties of the proposed PINNs approach are analyzed rigorously for the presented two-phase flow model by employing the Reynolds transport theorem in evolving domains, moreover, a comprehensive error estimation is provided to account for additional complexities introduced by the moving interface and the coupling between fluid dynamics and interface evolution. Numerical experiments are carried out to illustrate the effectiveness of the proposed PINNs approach for various configurations of two-phase flow moving interface problems, and to validate the theoretical findings as well. A practical guidance is thus provided for an efficient training set distribution when applying the proposed PINNs approach to two-phase flow moving interface problems in practice.

Figures (11)

• Figure 1: Two schematic evolving domain decompositions divided by the time-dependent interface $\Gamma(t)$: the immersed case (left) and the back-to-back case (right).
• Figure 2: The training set of Example 1 separated by a prescribed generalized helicoid with a deforming and translating elliptical profile curve. The elliptical interfaces that are projections of the helicoid on $xy$-plane at different times are plotted with different colors, while their interiors are uniformly covered by red dashed line segments at different angles $2\pi t$ along the time axis.
• Figure 3: PINNs outputs of Example 1 on the terminal planes of the outer and inner subdomains, respectively. Left: The horizontal velocity $u$; Middle: the vertical velocity $v$; Right: the pressure $p$; Upper: outside the elliptical interface; Lower: inside the elliptical interface.
• Figure 4: Error distributions between PINNs output and exact solution of Example 1 on the terminal planes of the outer and inner subdomains, respectively. Left: The horizontal velocity $u$; Middle: the vertical velocity $v$; Right: the pressure $p$; Upper: outside the elliptical interface; Lower: inside the elliptical interface.
• Figure 5: The training set of Example 2 separated by a prescribed generalized helicoid with a deforming and translating five-fold-modulation cyclic-harmonic profile curve. The cyclic-harmonic curves with a five-fold modulation, which are projections of the helicoid onto $xy$-planes at different times, are plotted with different colors, while their interiors are uniformly covered by red dashed line segments at different angles $2\pi t$ along the time axis.

Theorems & Definitions (2)

• Theorem 4.1
• Remark 5.1