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.

翻译：本文提出一种间断与尖点捕捉物理信息神经网络（PINN），用于求解含分段常数黏性及沿界面奇异力的Stokes方程。我们首先在各流体域内分别重构控制方程，并将奇异力效应替换为界面两侧解之间的牵引力平衡方程。由于压力在界面处存在间断、速度导数亦不连续，本文采用两个全连接子网络分别逼近压力与速度场。两个子网络共享相同的原始坐标输入参数，但采用不同的增强特征输入。这些增强输入包含界面信息——我们假定给定水平集函数，其零水平集指示界面位置。压力子网络采用指示函数作为增强输入以捕捉函数间断性，而速度子网络通过牵引力平衡方程采用尖点强制水平集函数来捕捉导数不连续性。通过二维与三维Stokes界面问题的数值实验，本文与文献中的增强浸入界面方法进行精度对比。结果表明，即使采用中等规模神经元与充足训练数据点的浅层网络，其预测精度也可达到与浸入界面方法相当的水平。