BENO: Boundary-embedded Neural Operators for Elliptic PDEs

• URL: http://arxiv.org/abs/2401.09323v1
• Date: Wed, 17 Jan 2024 16:47:39 GMT
• Title: BENO: Boundary-embedded Neural Operators for Elliptic PDEs
• Authors: Haixin Wang, Jiaxin Li, Anubhav Dwivedi, Kentaro Hara, Tailin Wu
• Abstract summary: We introduce Boundary-Embedded Neural Operators (BENO) for solving elliptic PDEs. BENO embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Our model outperforms state-of-the-art neural operators and strong baselines by an average of 60.96%.
• Score: 15.18712698704595
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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