Koopman operators are infinite-dimensional operators that linearize nonlinear dynamical systems, facilitating the study of their spectral properties and enabling the prediction of the time evolution of observable quantities. Recent methods have aimed to approximate Koopman operators while preserving key structures. However, approximating Koopman operators typically requires a dictionary of observables to capture the system's behavior in a finite-dimensional subspace. The selection of these functions is often heuristic, may result in the loss of spectral information, and can severely complicate structure preservation. This paper introduces Multiplicative Dynamic Mode Decomposition (MultDMD), which enforces the multiplicative structure inherent in the Koopman operator within its finite-dimensional approximation. Leveraging this multiplicative property, we guide the selection of observables and define a constrained optimization problem for the matrix approximation, which can be efficiently solved. MultDMD presents a structured approach to finite-dimensional approximations and can more accurately reflect the spectral properties of the Koopman operator. We elaborate on the theoretical framework of MultDMD, detailing its formulation, optimization strategy, and convergence properties. The efficacy of MultDMD is demonstrated through several examples, including the nonlinear pendulum, the Lorenz system, and fluid dynamics data, where we demonstrate its remarkable robustness to noise.

翻译：库普曼算子是无限维算子，可将非线性动力系统线性化，有助于研究其谱特性并实现可观测量时间演化的预测。近期方法致力于在保持关键结构的同时逼近库普曼算子。然而，逼近库普曼算子通常需要一组可观测量字典，以在有限维子空间中捕捉系统行为。这些函数的选择常依赖启发式方法，可能导致谱信息丢失，并严重增加结构保持的复杂性。本文提出乘法动态模式分解（MultDMD），该方法在其有限维逼近中强制保留了库普曼算子固有的乘法结构。通过利用这一乘法性质，我们指导可观测量选择，并定义了一个可高效求解的矩阵近似约束优化问题。MultDMD为有限维逼近提供了一种结构化方法，能更精确地反映库普曼算子的谱特性。我们详细阐述了MultDMD的理论框架，包括其公式化表达、优化策略和收敛性质。通过非线性摆、洛伦兹系统及流体动力学数据等多个实例，验证了MultDMD的有效性，并展示了其卓越的鲁棒性。