Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks

TL;DR

The paper addresses efficient, generalizable solving of static boundary value problems by learning a solution operator with graph neural networks trained on FEM-derived triangulations. By representing FEM meshes as graphs with node features and edge distances, the model learns to predict PDE solutions and derived quantities such as potentials and fields, generalizing across unseen geometries and superpositions of inhomogeneities. A key contribution is the emphasis on mesh augmentation and dataset diversity to achieve robust shape generalization, along with a public dataset and runtime improvements that outperform standard FEM in speed, especially on GPU. While potentials are accurately approximated, vector fields pose greater challenges due to physical constraint violations, indicating future work on enforcing physics-informed constraints and extending to broader PDE classes.

Abstract

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.

Figures (6)

• Figure 1: We train a neural network to predict solutions of boundary value problems and additional quantities like in this case the electric potential and electric field of an electrostatics simulation. Here we compare predictions of the model (pred) with ground truth data from an FEM simulation (gt). For the electric field, we visualize the magnitude of the field and overlay it with arrows depicting the field orientation.
• Figure 2: Visualized input and output quantities for the square mesh. For the scalar fields (potentials) we plot the magnitude. For the vector fields, we visualize the magnitude of the field and overlay it with arrows depicting the field orientation. (a) The charge input for the electrostatics problem. (b--c) Prediction targets in form of the potential and the electric field. (d) Current input for the magnetostatics problem. (e--f) Prediction targets in form of the magnetic vector potential (only $z$ component) and the magnetic vector field.
• Figure 3: Examples of the different mesh geometries used for solving the PDEs and constructing the dataset. The augmentations are applied to (i) the disk mesh and square mesh, where we vary the node density (ii) the disk with a hole, where we vary the hole size and location (iii) the L-mesh, where we control the size of the cutout. The nodes where Dirichlet boundary conditions are active are marked in green. Note that the different meshes also vary in resolution.
• Figure 4: Visualizations of the largest testing errors for our best performing models. We compare predictions of the model (pred) with ground truth data from the FEM simulation (gt). The background color depicts the magnitude of the respective field and the orientation is denoted by the arrows. The mean squared errors (vector field) for the shape experiments are $7.327\mathrm{E}{-3}$ (electrostatics) and $10.05\mathrm{E}{-3}$ (magnetostatics). The mean squared errors (vector field) for the superposition experiments are $1.013\mathrm{E}{-3}$ (electrostatics) and $1.154\mathrm{E}{-3}$ (magnetostatics).
• Figure 5: Runtime comparison of our GNN with the fenics FEM solver for predicting solutions on square meshes. We plot the time for solving a single example PDE versus the number of nodes in the mesh using a logarithmic scale. We compute each point on the curves as an average over 5 independent runs. The error bars depict the 95% confidence interval.