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

• URL: http://arxiv.org/abs/2301.10022v2
• Date: Mon, 6 May 2024 06:55:23 GMT
• Title: Koopman neural operator as a mesh-free solver of non-linear partial differential equations
• Authors: Wei Xiong, Xiaomeng Huang, Ziyang Zhang, Ruixuan Deng, Pei Sun, Yang Tian,
• Abstract summary: We propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, we can equivalently learn the solution of a non-linear PDE family. The KNO exhibits notable advantages compared with previous state-of-the-art models.
• Score: 15.410070455154138
• License: http://creativecommons.org/licenses/by/4.0/
• 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).

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