A Physics-driven GraphSAGE Method for Physical Process Simulations Described by Partial Differential Equations

TL;DR

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 the experiments.

Abstract

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.

Figures (11)

• Figure 1: The unstructured grid is represented as a graph (node 0 is the target node for simplicity). Columns in the nodal feature matrix $\mathbf{f}$ of the input graph represent nodal $x$-coordinates $\boldsymbol{X}_\text{coor}$, $y$-coordinates $\boldsymbol{Y}_\text{coor}$, and the input parameter vector $\boldsymbol{\mu}$ (e.g. the source terms), respectively. Each item in the edges feature vector $\boldsymbol{e}$ is a distance-related value. Each column in the adjacent list $\mathbf{A}$ denotes the directional connection between nodes of the first row to the second row.
• Figure 2: An example of updating nodal features of node 0 in the $k$th GNN layer. It involves three steps: message passing, message aggregation, and feature updating, enabling the target node 0 to update its features from $\boldsymbol{f}_{0}^{(k-1)}$ to $\boldsymbol{f}_{0}^{(k)}$.
• Figure 3: The training schematics of the proposed PD-GraphSAGE.
• Figure 4: Case 1: an electrostatic field problem with corner singularity. a) The mesh with local refinement, b) the numerical solution calculated by PD-GraphSAGE, and c) the reference solution calculated by FEM.
• Figure 5: Singularity area results and training comparison with different parameter settings of case 1. a) The comparison of calculated curves at $y=0$, b) the relative $L^2$ error of training PD-GraphSAGE separately with different $\epsilon$ of $E_{j,i}^d$ and $E_{j,i}=1$, and c) the relative $L^2$ error of training PD-GraphSAGE with different $L_\text{max}$ of $E_{j,i}^d$, where the largest element edge length of mesh is 0.1484 m $\textless$ 0.15 m.