A discontinuity and cusp capturing PINN for Stokes interface problems with discontinuous viscosity and singular forces

TL;DR

This paper introduces a specialized physics-informed neural network designed to accurately solve Stokes interface problems with discontinuous viscosity and singular forces, effectively capturing discontinuities and cusps at interfaces.

Contribution

It proposes a novel PINN architecture with dual sub-networks and augmented inputs to handle discontinuities and cusps in Stokes interface problems, improving accuracy over existing methods.

Findings

Achieves comparable accuracy to immersed interface methods.

Effectively captures pressure and velocity discontinuities.

Works for both 2D and 3D Stokes interface problems.

Abstract

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.