On the locality of local neural operator in learning fluid dynamics

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.

Learning Transient Partial Differential Equations with Local Neural Operators

In decades, enormous computational resources are poured into solving the transient partial differential equations for multifarious physical fields. The latest artificial intelligence has shown great potential in accelerating these computations, but its road to wide applications is hindered by the variety of computational domains and boundary conditions. Here, we overcome this obstacle by constructing a learning framework capable of purely representing the transient PDEs with local neural operators (LNOs). This framework is demonstrated in learning several transient PDEs, especially the Navier-Stokes equations, and successfully applied to solve problems with quite different domains and boundaries, including the internal flow, the external flow, and remarkably, the flow across the cascade of airfoils. In these applications, our LNOs are faster than the conventional numerical solver by over 1000 times, which could be significant for scientific computations and engineering simulations.