  In this paper, we present a discontinuity and cusp capturing physics-informed
neural network (PINN) to solve Stokes equations with a piecewise-constant
viscosity and singular force along an interface. We first reformulate the
governing equations in each fluid domain separately and replace the singular
force effect with the traction balance equation between solutions in two sides
along the interface. Since the pressure is discontinuous and the velocity has
discontinuous derivatives across the interface, we hereby use a network
consisting of two fully-connected sub-networks that approximate the pressure
and velocity, respectively. The two sub-networks share the same primary
coordinate input arguments but with different augmented feature inputs. These
two augmented inputs provide the interface information, so we assume that a
level set function is given and its zero level set indicates the position of
the interface. The pressure sub-network uses an indicator function as an
augmented input to capture the function discontinuity, while the velocity
sub-network uses a cusp-enforced level set function to capture the derivative
discontinuities via the traction balance equation. We perform a series of
numerical experiments to solve two- and three-dimensional Stokes interface
problems and perform an accuracy comparison with the augmented immersed
interface methods in literature. Our results indicate that even a shallow
network with a moderate number of neurons and sufficient training data points
can achieve prediction accuracy comparable to that of immersed interface
methods.
