  The lacking of analytic solutions of diverse partial differential equations
(PDEs) gives birth to a series of computational techniques for numerical
solutions. Although numerous latest advances are accomplished in developing
neural operators, a kind of neural-network-based PDE solver, these solvers
become less accurate and explainable while learning long-term behaviors of
non-linear PDE families. In this paper, we propose the Koopman neural operator
(KNO), a new neural operator, to overcome these challenges. With the same
objective of learning an infinite-dimensional mapping between Banach spaces
that serves as the solution operator of the target PDE family, our approach
differs from existing models by formulating a non-linear dynamic system of
equation solution. By approximating the Koopman operator, an
infinite-dimensional operator governing all possible observations of the
dynamic system, to act on the flow mapping of the dynamic system, we can
equivalently learn the solution of a non-linear PDE family by solving simple
linear prediction problems. We validate the KNO in mesh-independent, long-term,
and5zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes
equation and the Rayleigh-B{\'e}nard convection) and three real dynamic systems
(e.g., global water vapor patterns and western boundary currents). In these
experiments, the KNO exhibits notable advantages compared with previous
state-of-the-art models, suggesting the potential of the KNO in supporting
diverse science and engineering applications (e.g., PDE solving, turbulence
modelling, and precipitation forecasting).
