KoopmanLab: machine learning for solving complex physics equations
- URL: http://arxiv.org/abs/2301.01104v3
- Date: Sun, 19 Mar 2023 13:44:49 GMT
- Title: KoopmanLab: machine learning for solving complex physics equations
- Authors: Wei Xiong, Muyuan Ma, Xiaomeng Huang, Ziyang Zhang, Pei Sun, Yang Tian
- Abstract summary: We present KoopmanLab, an efficient module of the Koopman neural operator family, for learning PDEs without analytic solutions or closed forms.
Our module consists of multiple variants of the Koopman neural operator (KNO), a kind of mesh-independent neural-network-based PDE solvers.
The compact variants of KNO can accurately solve PDEs with small model sizes while the large variants of KNO are more competitive in predicting highly complicated dynamic systems.
- Score: 7.815723299913228
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Numerous physics theories are rooted in partial differential equations
(PDEs). However, the increasingly intricate physics equations, especially those
that lack analytic solutions or closed forms, have impeded the further
development of physics. Computationally solving PDEs by classic numerical
approaches suffers from the trade-off between accuracy and efficiency and is
not applicable to the empirical data generated by unknown latent PDEs. To
overcome this challenge, we present KoopmanLab, an efficient module of the
Koopman neural operator family, for learning PDEs without analytic solutions or
closed forms. Our module consists of multiple variants of the Koopman neural
operator (KNO), a kind of mesh-independent neural-network-based PDE solvers
developed following dynamic system theory. The compact variants of KNO can
accurately solve PDEs with small model sizes while the large variants of KNO
are more competitive in predicting highly complicated dynamic systems govern by
unknown, high-dimensional, and non-linear PDEs. All variants are validated by
mesh-independent and long-term prediction experiments implemented on
representative PDEs (e.g., the Navier-Stokes equation and the Bateman-Burgers
equation in fluid mechanics) and ERA5 (i.e., one of the largest high-resolution
global-scale climate data sets in earth physics). These demonstrations suggest
the potential of KoopmanLab to be a fundamental tool in diverse physics studies
related to equations or dynamic systems.
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