BENO: Boundary-embedded Neural Operators for Elliptic PDEs

Haixin Wang, Jiaxin LI, Anubhav Dwivedi, Kentaro Hara, Tailin Wu

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Keywords: AI for PDEs; physical simulation, neural operators, boundary-embedded

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2024/AuthorGuide.

TL;DR: We introduce a boundary-embedded neural operator that incorporates complex boundary shape and inhomogeneous boundary values into the solving of Elliptic PDEs

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 neglect 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 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 and strong baselines extensively in elliptic PDEs with complex 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%.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors' identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 5809