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

Wei Fan Hu, Yi Jun Shih, Te Sheng Lin, Ming Chih Lai

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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.

Original languageEnglish
Article number116486
JournalComputer Methods in Applied Mechanics and Engineering
Volume418
DOIs
StatePublished - 1 Jan 2024

Keywords

  • Evolving surfaces
  • Laplace–Beltrami operator
  • Physics-informed neural networks
  • Shallow neural network
  • Surface partial differential equations

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