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).

翻译：各类偏微分方程解析解的缺失催生了一系列数值求解的计算技术。尽管近年基于神经网络的偏微分方程求解器（即神经算子）取得了诸多进展，但在学习非线性偏微分方程族的长期行为时，这类求解器的精度与可解释性均有所下降。本文提出了一种新型神经算子——库普曼神经算子（Koopman neural operator, KNO），以克服上述挑战。与现有模型不同，KNO旨在学习巴拿赫空间之间无限维映射（即目标偏微分方程族的解算子）的同时，将方程解重构为非线性动力系统。通过逼近库普曼算子（描述该动力系统所有可能观测的无限维算子）并作用于系统流映射，我们可以将非线性偏微分方程族的求解等价转化为简单线性预测问题。我们在五个代表性偏微分方程（如纳维-斯托克斯方程和瑞利-贝纳德对流）及三个真实动力系统（如全球水汽分布和西边界流）上，验证了KNO的网格无关性、长期预测能力及零样本预测性能。实验结果表明，与先前最先进模型相比，KNO展现出显著优势，揭示了其在科学工程应用（如偏微分方程求解、湍流建模和降水预报）中的广阔潜力。