An Implicit GNN Solver for Poisson-like problems

TL;DR

Ψ-GNN presents a physics-informed, implicit GNN solver for Poisson-like problems on unstructured meshes with mixed boundary conditions. By embedding the FEM discretization into a graph framework and solving a fixed-point equation via a root-finding approach, the model automatically adapts the required message-passing depth and enforces boundary conditions by design, with stability guarantees from a spectral-radius regularization. Theoretical results establish a universal approximation property and fixed-point equivalence, while experiments show robust generalization to varying mesh sizes and boundary configurations, often with far fewer parameters than competing methods. The approach offers a promising path for integrating ML-based solvers into industrial CFD pipelines, with potential extensions to 3D and more complex PDEs.

Abstract

This paper presents $Ψ$-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems with mixed boundary conditions. By leveraging the Implicit Layer Theory, $Ψ$-GNN models an "infinitely" deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical prerequisite for physical applications, and is able to adapt to any initially provided solution. $Ψ$-GNN is trained using a "physics-informed" loss, and the training process is stable by design, and insensitive to its initialization. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, $Ψ$-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

Figures (11)

• Figure 1: (\ref{['fig:sparsenobounds']}) illustrates the sparsity pattern in matrix $A$, obtained by discretizing the Laplace operator in \ref{['poisson-equation']} using FEM for a problem with 17 nodes before applying Dirichlet boundary conditions. Using this matrix as an adjacency matrix, the induced graph is shown in (\ref{['fig:graphsketchnobound']}), resulting in a fully undirected graph. (\ref{['fig:sparsebounds']}) displays the sparsity pattern of the same matrix $A$after applying Dirichlet boundary conditions. The related graph is shown in (\ref{['fig:graphsketchbound']}), resulting in an undirected graph for Interior and Neumann nodes (blue and yellow nodes) and a directed graph for Dirichlet nodes (red nodes). For both (\ref{['fig:sparsenobounds']}) and (\ref{['fig:sparsebounds']}), Interior, Neumann and Dirichlet connections correspond to the blue, yellow and red squares, respectively.
• Figure 2: Diagram of $\Psi$-GNN: The model uses an Encode-Process-Decode architecture. The encoder maps an initial solution $U^0$ to some latent representation $H^0$. The processor outputs a final latent state $\widehat{H}$ by considering a different treatment for each node type. Dirichlet nodes are preserved during the process, and specific MPNN (red arrows) for Interior and Neumann nodes are computed to build a GNN function $h_\theta$. A black-box "root-finding" solver automatically propagates the information through the graph by finding the fixed point $\widehat{H}$ of $h_\theta$, starting from the initial guess $H^0$. The decoder maps $\widehat{H}$ back to the physical space to get the final solution $\widehat{U}$.
• Figure 3: Process of updating the Interior latent variable $z_{i}^\texttt{I}$: Firstly, two MPNNs, $\Phi_{\rightarrow,\theta}^\texttt{I}$ (red arrows) and $\Phi_{\leftarrow,\theta}^\texttt{I}$ (green arrows), are computed to account for the bi-directionality of the edges. These computed messages are then concatenated with problem-specific data $b_i$ and the actual latent state $H_i$ and passed through a trainable function $\Lambda_{i,\theta}$. The output of this trainable function is finally used to calculate $z_{i}^\texttt{I}$ in a Res-Net fashion.
• Figure 4: Geometries extracted from the synthetic dataset colored with respect to their node type (blue: Interior, red: Dirichlet, yellow: Neumann). These geometries are used to solve Poisson problems with Dirichlet boundary conditions only (on the left side) and mixed boundary conditions (on the right side). The arrows on the boundary of the two domains on the right represent the normal vectors.
• Figure 5: Figures \ref{['fig:forcingfunction']} and \ref{['fig:boundaryfunction']} display the discrete values of a forcing term $f$ and a boundary function $g$, on a mesh sampled from the test set. The coefficients of $f$ and $g$, uniformly sampled in $[-10, 10]$, are $r_1 = 3.2$, $r_2 = -7.5$, $r_3 = 1.1$, $r_4 = 5.7$, $r_5 = -9.5$, $r_6 = 0.47$, $r_7 = -8.8$, $r_8 = 9.11$, and $r_9 = 3.5$. The considered mesh is shown as a grey shadow in the plot.

Theorems & Definitions (10)

• Proposition 1: Equivalence of direct and fixed-point formulations
• proof
• Proposition 2: Satisfying the hypotheses of Corollary 1 in DSS
• proof
• Corollary 1: Existence of a GNN model approximating $\varphi$
• Theorem 1: Universal Approximation Property
• proof
• Proposition 3: Contractivity of $h_{\theta\varepsilon}$
• proof
• Lemma 1: Continuity of $\varphi$