MESH-INDEPENDENT OPERATOR LEARNING FOR PDES USING SET REPRESENTATIONS

Abstract

Operator learning, which learns the mapping between infinite-dimensional function spaces, is an attractive alternative to traditional numerical methods for solving partial differential equations (PDEs). In practice, the functions of the physical systems are often observed by sparse or even irregularly distributed measurements; thus, the functions are discretized and usually represented by finite structured arrays, which are given as data of input-output pairs. Through training with the arrays, the solution of the trained models should be independent of the discretization of the input function and can be queried at any point continuously. Therefore, the architectures for operator learning should be flexibly compatible with arbitrary sizes and locations of the measurements, otherwise, scalability can be restricted when the observations have discrepancies between measurement formats. In this study, we propose the proper treatment of discretized functions as set-valued data and construct an attention-based model, called mesh-independent operator learner (MIOL), to provide proper treatments of input functions and query coordinates for the solution functions by detaching the dependence of the input and output meshes. Our models pre-trained with benchmark datasets of operator learning are evaluated by downstream tasks to demonstrate the generalization abilities to varying discretization formats of the system, which are natural characteristics of the continuous solution of the PDEs.

1. INTRODUCTION

Partial Differential equations (PDEs) are among the most successful mathematical tools for representing the physical systems with governing equations over infinitesimal segments of the domain of interest, given some problem-specific boundary conditions or forcing functions (Mizohata, 1973) . The governing PDEs, which are globally shared in the entire domain, are interpreted as interactions between infinitesimal segments with respect to their geometrical structures and values. Because of the universality of the entire domain, the system can be analyzed in a continuous manner with respect to the system inputs and outputs. In general, identifying appropriate governing equations for unknown systems is very challenging without domain expertise, however, numerous unknown processes remain for many complex systems. Even if knowing the governing equation of the system is known, it requires unnecessary time and memory costs to be solved using conventional numerical methods, and sometimes it is intractable to compute in a complex and large-scale system. Motivation. In recent years, operator learning, an alternative to conventional numerical methods, has been gaining attention, pursuing mapping between infinite-dimensional input/output function spaces in a data-driven manner without any problem-specific knowledge of the system (Nelsen & Stuart, 2021; Li et al., 2020a; b; 2021b; Lu et al., 2019; 2021; Cao, 2021; Kovachki et al., 2021) . Intuitively, for the underlying PDE, L a u = f defined on the continuous bounded domains Ω with system parameters a ∈ A, forcing function f ∈ F, and the solution of the system u ∈ U, the goal of the operator learning is to approximate the inverse operator G = L -1 a f : A → U or G : F → U with parametric model G θ . Without loss of generality, when the input function is a, the output function can be computed as u = G θ (a). Because the operator G θ should be able to capture interactions between elements of system inputs a to discover the governing PDEs, G θ is approximated by a series of integral operators with parameterized kernels that iteratively update the system input to the output (Nelsen & Stuart, 2021; Li et al., 2020a; b; 2021b; Cao, 2021; Kovachki et al., 2021) . In practice,

