@Article{AAM-39-4,
author = {Yu-Hau, Tseng and Ming-Chih, Lai},
title = {A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces},
journal = {Annals of Applied Mathematics},
year = {2023},
volume = {39},
number = {4},
pages = {385--405},
abstract = {<p style="text-align: justify;">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&nbsp;sufficient training data points can achieve prediction accuracy comparable to
that of immersed interface methods.</p>},
issn = {},
doi = {https://doi.org/10.4208/aam.OA-2023-0028},
url = {https://global-sci.com/article/72620/a-discontinuity-and-cusp-capturing-pinn-for-stokes-interface-problems-with-discontinuous-viscosity-and-singular-forces}
}