GUIDING CONTINUOUS OPERATOR LEARNING THROUGH PHYSICS-BASED BOUNDARY CONSTRAINTS

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. 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 physicsbased 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).

1. INTRODUCTION

Partial differential equations (PDEs) are ubiquitous in many scientific and engineering applications. Often, these PDEs involve boundary value constraints, known as Boundary Conditions (BCs), in which certain values are imposed at the boundary of the domain where the solution is supposed to be obtained. Consider the heat equation that models heat transfer in a one dimensional domain as shown schematically in Figure 1 . The left and right boundaries are attached to an insulator (zero heat flux) and a heater (with known heat flux), respectively, which impose certain values for the derivatives of the temperature at the boundary points. No-slip boundary condition for wall-bounded viscous flows, and periodic boundary condition for modeling isotropic homogeneous turbulent flows are other examples of boundary constraints widely used in computational fluid dynamics. Violating these boundary constraints can lead to unstable models and non-physical solutions. Thus, it is critical for a PDE solver to satisfy these constraints in order to capture the underlying physics accurately, and provide reliable models for rigorous research and engineering design. In the context of solving PDEs, there has been an increasing effort in leveraging machine learning methods and specifically deep neural networks to overcome the challenges in conventional numerical methods (Adler & Öktem, 2017; Afshar et al., 2019; Guo et al., 2016; Khoo et al., 2020; Zhu & Zabaras, 2018) . One main stream of neural-network approaches has focused on training models that predict the solution function directly. These methods typically are tied to a specific resolution and PDE parameters, and may not generalize well to different settings. Many of these approaches may learn physical constraints implicitly through training data, and thus do not guarantee their satisfaction at test time (Greenfeld et al., 2019; Raissi et al., 2019; Wang et al., 2021) . Some previous works also attempt to formulate these physics-based constraints in the form of a hard constraint

