TY - JOUR

T1 - A shallow physics-informed neural network for solving partial differential equations on static and evolving surfaces

AU - Hu, Wei Fan

AU - Shih, Yi Jun

AU - Lin, Te Sheng

AU - Lai, Ming Chih

N1 - Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2024/1/1

Y1 - 2024/1/1

N2 - In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network (PINN) for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of a level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So, instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We demonstrate a series of performance study for the present methodology by solving Laplace–Beltrami equations and surface diffusion equations on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. We then extend the present methodology to solve the advection–diffusion equation on an evolving surface with a given velocity. To track the deforming surface, we additionally introduce a network, in which a prescribed hidden layer is employed to enforce the topological structure of the surface and learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate surfactant transportation on a droplet surface under shear flow and obtain some physically plausible results.

AB - In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network (PINN) for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of a level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So, instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We demonstrate a series of performance study for the present methodology by solving Laplace–Beltrami equations and surface diffusion equations on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. We then extend the present methodology to solve the advection–diffusion equation on an evolving surface with a given velocity. To track the deforming surface, we additionally introduce a network, in which a prescribed hidden layer is employed to enforce the topological structure of the surface and learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate surfactant transportation on a droplet surface under shear flow and obtain some physically plausible results.

KW - Evolving surfaces

KW - Laplace–Beltrami operator

KW - Physics-informed neural networks

KW - Shallow neural network

KW - Surface partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=85173056122&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2023.116486

DO - 10.1016/j.cma.2023.116486

M3 - 期刊論文

AN - SCOPUS:85173056122

SN - 0045-7825

VL - 418

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

M1 - 116486

ER -