A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces  

作　　者：Yu-Hau Tseng Ming-Chih Lai 

机构地区：[1]Department of Applied Mathematics,National University of Kaohsiung,Kaohsiung 81148,Taiwan [2]Department of Applied Mathematics,National Yang Ming Chiao Tung University,1001 Ta Hsueh Road,Hsinchu 300,Taiwan

出　　处：《Annals of Applied Mathematics》2023年第4期385-405,共21页应用数学年刊（英文版）

基　　金：supports by National Science and Technology Council,Taiwan,under research grants 111-2115-M-390-002 and 110-2115-M-A49-011-MY3,respectively.

摘　　要：In this paper,we present a discontinuity and cusp capturing physicsinformed neural network(PINN)to solve Stokes equations with a piecewiseconstant 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.

关 键 词：Stokes interface problem immersed interface method level set function physics-informed neural network discontinuity capturing shallow neural network cuspcapturing neural network 

分 类 号：O359[理学—流体力学] TP183[理学—力学]