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.