Guiding continuous operator learning through Physics-based boundary constraintsDownload PDF

Published: 01 Feb 2023, Last Modified: 13 Feb 2023ICLR 2023 posterReaders: Everyone
Keywords: partial differential equations, operator learning, physics-constraints, boundary conditions, kernel correction
Abstract: Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical meaning, and guarantee the existence and the uniqueness of the PDE solution. Current neural-network based approaches that aim to solve PDEs rely only on training data to help the model learn BCs implicitly, however, there is no guarantee of BC satisfaction by these models during evaluation. In this work, we propose Boundary enforcing Operator Network (BOON) that enables the BC satisfaction of neural operators by making structural changes to the operator kernel. We provide our refinement procedure, and demonstrate the satisfaction of physics-based BCs such as Dirichlet, Neumann, and periodic by the solutions obtained by BOON. Numerical experiments based on multiple PDEs with a wide variety of applications indicate that the proposed approach ensures satisfaction of BCs, and leads to more accurate solutions over the whole domain. The proposed method exhibits a (2X-20X) improvement in accuracy (0.000084 relative $L^2$ error for Burgers' equation). Code available at:
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.
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
Submission Guidelines: Yes
Please Choose The Closest Area That Your Submission Falls Into: Deep Learning and representational learning
TL;DR: We propose novel kernel correction mechanisms for neural operators to satisfy physical boundary constraints which are effective in improving the overall performance.
Supplementary Material: zip
18 Replies