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Area of Science:

  • Computational Science and Engineering
  • Applied Mathematics
  • Machine Learning

Background:

  • Complex physical systems often require solving time-dependent nonlinear partial differential equations (PDEs).
  • Full order models (FOMs) provide accuracy but are computationally intensive.
  • Surrogate models aim to balance accuracy and efficiency for faster simulations.

Purpose of the Study:

  • To explore the use of graph neural networks (GNNs) for simulating time-dependent PDEs with geometrical variability.
  • To develop a data-driven time-stepping surrogate model using GNNs.
  • To address the challenge of parameter-dependent spatial domains in PDE simulations.

Main Methods:

  • A GNN architecture is employed within a data-driven time-stepping scheme.
  • The approach is designed to handle parameter-dependent spatial domains and varying mesh resolutions.
  • Numerical experiments are conducted on both 2D and 3D problems.

Main Results:

  • The proposed GNN-based surrogate model demonstrates effectiveness in simulating time-dependent PDEs.
  • The approach successfully tackles problems with geometrical variability and different mesh resolutions.
  • GNNs show potential for generalization to new, unseen scenarios.

Conclusions:

  • Graph neural networks present a viable and efficient alternative to traditional surrogate models for simulating PDEs.
  • The GNN approach offers significant advantages in computational efficiency and generalization capabilities.
  • This method is particularly effective for problems involving geometrical variations.