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.

翻译：许多物理学理论扎根于偏微分方程（PDEs）。然而，日益复杂的物理方程，尤其是那些缺乏解析解或封闭形式的方程，阻碍了物理学的进一步发展。通过经典数值方法计算求解偏微分方程会在精度与效率之间进行权衡，且不适用于由未知潜在偏微分方程生成的经验数据。为克服这一挑战，我们提出了KoopmanLab，一个Koopman神经算子系列的高效模块，用于学习无解析解或封闭形式的偏微分方程。我们的模块包含Koopman神经算子（KNO）的多种变体，这是一种基于动力系统理论开发的无网格神经网络偏微分方程求解器。KNO的紧凑变体能够以较小的模型尺寸精确求解偏微分方程，而KNO的大规模变体在预测由未知、高维和非线性偏微分方程支配的高度复杂动力系统方面更具竞争力。所有变体均通过基于代表性偏微分方程（例如，流体力学中的Navier-Stokes方程和Bateman-Burgers方程）以及ERA5（地球物理学中最大的高分辨率全球气候数据集之一）实现的无网格和长期预测实验进行了验证。这些演示表明，KoopmanLab有潜力成为与方程或动力系统相关的多样化物理学研究中的基础工具。