Local neural operator for solving transient partial differential equations on varied domains

The neural operator method is reported to be able to rapidly solve partial differential equations (PDEs) in applied mechanics and engineering. However, the great efficiency is not fully realized in practice as the pre-trained models can hardly deal with the varied computational domain. In this work, we propose local neural operator (LNO) to approximate a local-related and shift-invariant time-marching operator when solving transient PDEs, in which the neural operators learn equations separated from case-specific conditions and thus make the domain flexible. It comes with a handy implementation including boundary treatments, enabling one pre-trained LNO to solve on varied domains. For demonstration, we let LNO learn typical transient PDEs, including viscous Burgers’ equation, wave equation, and Navier–Stokes (N–S) equation from randomly generated field samples. Then, the pre-trained LNO models solve demo problems on unseen domains different from training ones. Significantly, the pre-trained LNO solves practical flows (governed by N–S equation) three orders of magnitude faster than the conventional finite element method at most. This advantage of LNO is attributed to its pure explicit approximation for the large-interval time-marching operator.