On the locality of local neural operator in learning fluid dynamics

TL;DR

This paper investigates the importance of receptive range in local neural operators for fluid dynamics, showing that an appropriately chosen receptive range enhances accuracy and stability in solving PDEs.

Contribution

It provides a detailed analysis of how receptive range affects local neural operator performance and offers practical guidelines for designing effective LNO architectures.

Findings

Proper receptive range is crucial for LNO performance.

Over-small receptive range causes numerical oscillation.

Over-large receptive range reduces accuracy.

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.