Guiding continuous operator learning through Physics-based boundary constraints
This addresses the issue of unreliable PDE solutions for scientific computing and engineering applications, but it is incremental as it builds on existing neural operator methods.
The paper tackled the problem of neural-network based PDE solvers not guaranteeing boundary condition satisfaction, proposing BOON to enforce physics-based constraints, resulting in a 2X-20X improvement in relative L^2 error, such as 0.000084 for Burgers' equation.
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. 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, e.g. 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 entire domain. The proposed correction method exhibits a (2X-20X) improvement over a given operator model in relative $L^2$ error (0.000084 relative $L^2$ error for Burgers' equation).