A Physics-driven GraphSAGE Method for Physical Process Simulations Described by Partial Differential Equations
This work addresses computational physics challenges for researchers and engineers by providing a more efficient and accurate method for simulating irregular PDEs and creating parametric surrogate models, though it is incremental as it builds on existing GraphSAGE and PINN techniques.
The authors tackled the problem of low accuracy in physics-informed neural networks for irregular PDEs and the lack of parametric surrogate models, proposing a physics-driven GraphSAGE method that achieves accurate solutions and is several times faster than the finite element method in heat conduction experiments.
Physics-informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and oscillations, trained solutions usually suffer low accuracy. In addition, most current works only offer the trained solution for predetermined input parameters. If any change occurs in input parameters, transfer learning or retraining is required, and traditional numerical techniques also need an independent simulation. In this work, a physics-driven GraphSAGE approach (PD-GraphSAGE) based on the Galerkin method and piecewise polynomial nodal basis functions is presented to solve computational problems governed by irregular PDEs and to develop parametric PDE surrogate models. This approach employs graph representations of physical domains, thereby reducing the demands for evaluated points due to local refinement. A distance-related edge feature and a feature mapping strategy are devised to help training and convergence for singularity and oscillation situations, respectively. The merits of the proposed method are demonstrated through a couple of cases. Moreover, the robust PDE surrogate model for heat conduction problems parameterized by the Gaussian random field source is successfully established, which not only provides the solution accurately but is several times faster than the finite element method in our experiments.