ncku-talk2023-11-16

Colloquium

DATE 2023-11-16　16:10-17:00

PLACE 數學系館 1F3174教室

SPEAKER 曾昱豪教授（國立高雄大學應用數學系）

TITLE A Machine Learning Method for Solving Stokes Interface Problems

ABSTRACT In this talk, we present a physics-informed neural network called the Discontinuity-Cusps Capturing Physics-Informed Neural Network for solving piecewise-constant viscosity Stokes interface problems. The network consists of two fully connected sub-networks that handle the pressure and velocity vectors separately. These sub-networks share the same primary input arguments but have different augmented features: the Discontinuity-Capturing Shallow Neural Network (DCSNN) uses an indicator function to capture the discontinuities, while the Cusp-Capturing Neural Network (CuspNN) employs a cusp-enforced level set function to capture cusp-like velocity profiles caused by jumps in viscous stresses. The main objective of this study is to explore the use of the stress balance formulation directly in the training process for obtaining accurate predictions, as opposed to the force formulation used in the Immersed Interface Method (IIM). We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and demonstrate the effectiveness and accuracy of the proposed network model. 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.

Colloquium

DATE	2023-11-16　16:10-17:00

PLACE	數學系館 1F3174教室

SPEAKER	曾昱豪教授（國立高雄大學應用數學系）

TITLE	A Machine Learning Method for Solving Stokes Interface Problems

ABSTRACT	In this talk, we present a physics-informed neural network called the Discontinuity-Cusps Capturing Physics-Informed Neural Network for solving piecewise-constant viscosity Stokes interface problems. The network consists of two fully connected sub-networks that handle the pressure and velocity vectors separately. These sub-networks share the same primary input arguments but have different augmented features: the Discontinuity-Capturing Shallow Neural Network (DCSNN) uses an indicator function to capture the discontinuities, while the Cusp-Capturing Neural Network (CuspNN) employs a cusp-enforced level set function to capture cusp-like velocity profiles caused by jumps in viscous stresses. The main objective of this study is to explore the use of the stress balance formulation directly in the training process for obtaining accurate predictions, as opposed to the force formulation used in the Immersed Interface Method (IIM). We perform a series of numerical experiments to solve two- and three-dimensional Stokes interface problems and demonstrate the effectiveness and accuracy of the proposed network model. 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.