  In this paper, a meshfree method using the deep neural network (DNN) approach
is developed for solving two kinds of dynamic 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 to be studied 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 belongs
to fluid-structure interaction (FSI) problems modeled by Navier-Stokes
equations on one side of the interface and the structural equation on the other
side of the interface, both the fluid and the structure interact with each
other via the kinematic- and the dynamic interface conditions across the
interface. The DNN/meshfree method is respectively developed for the above
two-phase interface problems by representing solutions of PDEs using the DNNs'
structure and reformulating the dynamic interface problem as a least-squares
minimization problem based upon a space-time sampling point set. Approximation
error analyses are also carried out for each kind of interface problem, which
reveals an intrinsic strategy about how to efficiently build a sampling-point
training dataset to obtain a more accurate DNNs' approximation. In addition,
compared with traditional discretization approaches, the proposed DNN/meshfree
method and its error analysis technique can be smoothly extended to many other
dynamic interface problems with fixed interfaces. Numerical experiments are
conducted to illustrate the accuracies of the proposed DNN/meshfree method for
the presented two-phase interface problems. Theoretical results are validated
to some extent through three numerical examples.
