KoopmanLab: Machine learning for solving complex physics equations

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 (KNO) family, for learning PDEs without analytic solutions or closed forms. Our module consists of multiple variants of the KNO, a kind of mesh-independent neural-network-based PDE solvers developed following the 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 datasets 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.