Mesh-Independent Operator Learning for PDEs using Set Representations

Seungjun Lee

Keywords: partial differential equations, operator learning, set representations, attention-based model, implicit neural representation

TL;DR: We propose an attention-based operator learning model for obtaining the continuous solution of PDEs, independent of the discretization formats.

Abstract: Operator learning, learning the mapping between infinite-dimensional function spaces, has been attracted as an alternative approach to traditional numerical methods to solve 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, it can restrict the scalability when the observations have discrepancies between measurement formats. In this paper, we propose to treat the 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 dependencies on 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.

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