On the locality of local neural operator in learning fluid dynamics

TL;DR

The paper presents elsarticle.cls as a robust LaTeX document class tailored for Elsevier submissions, built on article.cls to minimize package conflicts and to support multiple formatting styles. It details the class's dependencies, compatibility with natbib and AMS packages, and its flexible front matter handling to streamline manuscript preparation. It contrasts elsarticle.cls with the older elsart.cls, highlighting improvements in compatibility, preprint and final formatting options, and structured author affiliations. The practical contribution lies in providing clear installation guidance via Elsevier resources and CTAN, enabling authors to produce consistent, publication-ready manuscripts with reduced formatting risks. Overall, the work enhances reliability and efficiency in preparing LaTeX submissions for Elsevier journals.

Abstract

This paper launches a thorough discussion on the locality of local neural operator (LNO), which is the core that enables LNO great flexibility on varied computational domains in solving transient partial differential equations (PDEs). We investigate the locality of LNO by looking into its receptive field and receptive range, carrying a main concern about how the locality acts in LNO training and applications. In a large group of LNO training experiments for learning fluid dynamics, it is found that an initial receptive range compatible with the learning task is crucial for LNO to perform well. On the one hand, an over-small receptive range is fatal and usually leads LNO to numerical oscillation; on the other hand, an over-large receptive range hinders LNO from achieving the best accuracy. We deem rules found in this paper general when applying LNO to learn and solve transient PDEs in diverse fields. Practical examples of applying the pre-trained LNOs in flow prediction are presented to confirm the findings further. Overall, with the architecture properly designed with a compatible receptive range, the pre-trained LNO shows commendable accuracy and efficiency in solving practical cases.