  Artificial intelligence (AI) shows great potential to reduce the huge cost of
solving partial differential equations (PDEs). However, it is not fully
realized in practice as neural networks are defined and trained on fixed
domains and boundaries. Herein, we propose local neural operator (LNO) for
solving transient PDEs on varied domains. It comes together with a handy
strategy including boundary treatments, enabling one pre-trained LNO to predict
solutions on different domains. For demonstration, LNO learns Navier-Stokes
equations from randomly generated data samples, and then the pre-trained LNO is
used as an explicit numerical time-marching scheme to solve the flow of fluid
on unseen domains, e.g., the flow in a lid-driven cavity and the flow across
the cascade of airfoils. It is about 1000$\times$ faster than the conventional
finite element method to calculate the flow across the cascade of airfoils. The
solving process with pre-trained LNO achieves great efficiency, with
significant potential to accelerate numerical calculations in practice.
