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Jin, X., Cai, S., Li, H. and Karniadakis, G.E. (2021) NSFnets (Navier-Stokes Flow Nets): Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations. Journal of Computational Physics, 426, Article ID: 109951. https://doi.org/10.1016/j.jcp.2020.109951

has been cited by the following article:

• TITLE: Physical-Informed Neural Networks (PINNs) for Solving Shape Optimization Problems

  AUTHORS: Huanyu Li, Xiaoyan Li, Fangying Song

  KEYWORDS: PINNs, Phase-Field, Shape Optimization, Incompressible Navier-Stokes Equations

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.12 No.10, October 31, 2024

  ABSTRACT: In this paper, we use Physics-Informed Neural Networks (PINNs) to solve shape optimization problems. These problems are based on incompressible Navier-Stokes equations and phase-field equations. The phase-field function is used to describe the state of the fluids, and the optimal shape optimization is obtained by using the shape sensitivity analysis based on the phase-field function. The sharp interface is also presented by a continuous function between zero and one with a large gradient. To avoid the numerical solutions falling into the trivial solution, the hard boundary condition is implemented for our PINNs’ training. Finally, numerical results are given to prove the feasibility and effectiveness of the proposed numerical method.