BENO: Boundary-embedded Neural Operators for Elliptic PDEs
Haixin Wang Jiaxin Li Anubhav Dwivedi Kentaro Hara Tailin Wu
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.