BENO: Boundary-embedded Neural Operators for Elliptic PDEs

TL;DR

This work tackles solving elliptic PDEs with complex boundary geometry and inhomogeneous boundary conditions by introducing Boundary-Embedded Neural Operators (BENO). BENO integrates dual Graph Neural Network branches for interior and boundary effects with a Transformer-based boundary encoder, inspired by Green's function, to inject global boundary information into every message passing layer. Experiments on diverse boundary shapes, values, and grid resolutions show BENO significantly surpasses state-of-the-art neural operators, with an average improvement of about $60.96\%$ in relative error and strong generalization to unseen boundaries and higher resolutions. The method advances practical PDE-solving in realistic domains and provides a scalable, boundary-aware surrogate for elliptic problems, with code available at the provided GitHub repository.

Abstract

Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural operators have emerged as a promising technique to solve elliptic PDEs more efficiently by directly mapping the input to solutions. However, existing networks typically cannot handle complex geometries and inhomogeneous boundary values present in the real world. Here we introduce Boundary-Embedded Neural Operators (BENO), a novel neural operator architecture that embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Inspired by classical Green's function, BENO consists of two branches of Graph Neural Networks (GNNs) for interior source term and boundary values, respectively. Furthermore, a Transformer encoder maps the global boundary geometry into a latent vector which influences each message passing layer of the GNNs. We test our model extensively in elliptic PDEs with various boundary conditions. We show that all existing baseline methods fail to learn the solution operator. In contrast, our model, endowed with boundary-embedded architecture, outperforms state-of-the-art neural operators and strong baselines by an average of 60.96\%. Our source code can be found https://github.com/AI4Science-WestlakeU/beno.git.

Figures (9)

• Figure 1: Examples of different geometries for the elliptic PDEs: (a) forcing terms and (b) solutions. The nodes in red-orange color-map represent the complex, inhomogeneous boundary values. The redder the area, the higher the boundary value it represents, whereas the more orange the area, the lower the boundary value.
• Figure 2: Visualization of the graph construction on our train/set samples from 5 different corner elliptic datasets. The interior nodes are in black and the boundary one in purple.
• Figure 3: Overall architecture of our proposed BENO. The pink branch corresponds to the first term in Eq. \ref{['eq:decouple']}, and the green branch corresponds to the second term. As the backbone of boundary embedding, Transformer provides boundary information as a supplement for BE-MPNN, thereby enabling better prediction under complex boundary geometry and inhomogeneous boundary values.
• Figure 4: Visualization of two samples' prediction and prediction error from 4-Corners dataset. We render the solution $u$ of the baseline MP-PDE, our BENO and the ground truth in $\Omega$.
• Figure 5: Visualization of the convergence curve of our BENO and two baselines.