Koopman neural operator as a mesh-free solver of non-linear partial differential equations

TL;DR

The paper introduces the Koopman neural operator (KNO), a mesh-free neural network approach that efficiently learns solutions to non-linear PDEs by approximating the Koopman operator, enabling accurate long-term predictions and broad scientific applications.

Contribution

The paper proposes the Koopman neural operator (KNO), a novel neural network that models non-linear PDE solutions through linear prediction of the Koopman operator, improving accuracy and explainability.

Findings

KNO achieves mesh-independent, long-term, zero-shot predictions on complex PDEs.

KNO outperforms previous models in accuracy and efficiency.

KNO demonstrates versatility in real dynamic systems like water vapor patterns.

Abstract

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).