Physics-Informed Neural Networks for Solving Dynamic Two-Phase Interface Problems

.In this paper, based on the physics-informed neural networks (PINNs) framework, a meshfree method using the deep neural network approach is developed for solving two kinds of two-phase interface problems governed by different dynamic partial differential equations on either side of the stationary interface with the jump and high-contrast coefficients. The first type of two-phase interface problem is the fluid-fluid (two-phase flow) interface problem modeled by Navier–Stokes equations with high-contrast physical parameters across the interface. The second one is the fluid-structure interaction problem modeled by Navier–Stokes equations on one side of the interface and the structural equation on the other side, where the fluid and the structure interact with each other via the kinematic and dynamic interface conditions across the interface. Following the PINNs framework, the DNN/meshfree method is respectively developed for two kinds of two-phase interface problems by approximating the solutions using different DNN's structures in different subdomains and reformulating the interface problems as least-squares minimization problems based on a space-time sampling-point set (as the training dataset). Mathematically, the approximation error analyses are carried out for both interface problems, revealing an intrinsic strategy for efficiently sampling points to improve the accuracy. In addition, compared with traditional discretization approaches (e.g., finite element/volume/difference methods), the proposed DNN/meshfree method and its error analysis technique can be smoothly extended to many other dynamic interface problems with stationary interfaces. Numerical experiments illustrate the accuracy of the proposed method for the presented two-phase interface problems and validate theoretical results to some extent through two numerical examples.Keywordsdeep neural network (DNN)physics-informed neural networks (PINNs)two-phase flow interface problemfluid-structure interaction problem (FSI)meshfree methodleast-squares (LS) loss functionalapproximation accuracyMSC codes35Q3065M1568Q3274F1082C32