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 1 ). Violation of the left boundary constraint by an existing neural operator (FNO), even when trained on data satisfying boundary constraint, suggests heat flowing across the insulator, which is not aligned with the underlying physics. Disagreement at the right boundary violates energy conservation. The proposed boundary corrected model (BOON) produces physically relevant solution along with better overall accuracy (see Section 4.2.1). optimization problem, although they could be computationally expensive, and may not guarantee model convergence to more accurate results (Xu & Darve, 2020; Krishnapriyan et al., 2021; Lu et al., 2021b; Donti et al., 2021) . Neural Operators (NOs) are another stream of research which we pursue in our study here that aim to learn the operator map without having knowledge of the underlying PDEs (Li et al., 2020a; Rackauckas et al., 2020; Tran et al., 2021; Li et al., 2020b; Guibas et al., 2021; Bhattacharya et al., 2021; Gupta et al., 2021) . NO-based models can be invariant to PDE discretization resolution, and are able to transfer solutions between different resolutions. Despite the advantages NO-based models offer, they do not yet guarantee the satisfaction of boundary constraints. In Figure 1 , the variation of heat flux is shown at the boundary regions for a vanilla NO-based model (FNO) and our proposed model (BOON). FNO violates both boundary constraints, and results in a solution which deviates from the system underlying physics. Physics-based boundary conditions are inherent elements of the problem formulation, and are readily available. These constraints are mathematically well-defined, and can be explicitly utilized in the structure of the neural operator. We show that careful leverage of this easily available information improves the overall performance of the neural operator. We propose Boundary enforcing Operator Network (BOON), which allows for the use of this physical information. Given an integral kernelbased NO representing a PDE solution, a training dataset D, and a prescribed BC, BOON applies structural corrections to the neural operator to ensure the BC satisfaction by the predicted solution. Our main contributions in this work can be summarized as follows: (i) A systematic change to the kernel architecture to guarantee BC satisfaction. (ii) Three numerically efficient algorithms to implement BC correction in linear space complexity and no increase in the cost complexity of the given NO while maintaining its resolution-independent property. (iii) Proof of error estimates to show bounded changes in the solution. (iv) Experimental results demonstrating that our proposed BOON has state-of-the-art performance on a variety of physically-relevant canonical problems ranging from 1D time-varying Burgers' equation to complex 2D time-varying nonlinear Navier-Stokes lid cavity problem with different BC, e.g. Dirichlet, Neumann, and periodic.

2. TECHNICAL BACKGROUND

Section 2.1 formally describes the boundary value problem. Then, we briefly describe the operators in Sections 2.2 and 2.3, which are required for developing our solution in Section 3.

2.1. BOUNDARY VALUE PROBLEMS

Boundary value problems (BVPs) are partial differential equations (PDEs), in which the solutions are required to satisfy a set of constraints along given spatial locations. Primarily, the constraints assign physical meaning, and are required for uniqueness of the solution. Formally, a BVP is written as:



Heat equation with physical boundary constraints. Heat flow across a conductor with an insulator and a heater at the left and right boundary, respectively. The insulator physically enforces zero heat flux (∝ ∂u ∂x ), and the heater imposes a known heat flux α N (1, t) from the right boundary. Neumann denotes this derivative-based boundary constraint (see Table

