Koopman neural operator as a mesh-free solver of non-linear partial differential equations
This addresses the problem of improving accuracy and explainability in neural-network-based PDE solvers for science and engineering applications, representing a novel method rather than an incremental improvement.
The paper tackles the challenge of learning long-term behaviors of non-linear PDE families with neural operators, which often become less accurate and explainable, by proposing the Koopman neural operator (KNO) that learns solution operators through linear prediction problems. It validates KNO on five PDEs and three real dynamic systems, showing notable advantages over previous state-of-the-art models in mesh-independent, long-term, and zero-shot predictions.
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{é}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).