Data-driven neural Koopman operator theory has emerged as a powerful tool for linearizing and controlling nonlinear robotic systems. However, the performance of these data-driven models fundamentally depends on the trade-off between sample size and model dimensions, a relationship for which the scaling laws have remained unclear. This paper establishes a rigorous framework to address this challenge by deriving and empirically validating scaling laws that connect sample size, latent space dimension, and downstream control quality. We derive a theoretical upper bound on the Koopman approximation error, explicitly decomposing it into sampling error and projection error. We show that these terms decay at specific rates relative to dataset size and latent dimension, providing a rigorous basis for the scaling law. Based on the theoretical results, we introduce two lightweight regularizers for the neural Koopman operator: a covariance loss to help stabilize the learned latent features and an inverse control loss to ensure the model aligns with physical actuation. The results from systematic experiments across six robotic environments confirm that model fitting error follows the derived scaling laws, and the regularizers improve dynamic model fitting fidelity, with enhanced closed-loop control performance. Together, our results provide a simple recipe for allocating effort between data collection and model capacity when learning Koopman dynamics for control.

翻译：数据驱动的神经Koopman算子理论已成为线性化与控制非线性机器人系统的有力工具。然而，这些数据驱动模型的性能根本上取决于样本量与模型维度之间的权衡关系，其缩放规律一直不明确。本文通过推导并实证验证连接样本量、潜在空间维度与下游控制质量的缩放定律，建立了一个严谨的框架以应对这一挑战。我们推导了Koopman近似误差的理论上界，将其显式分解为采样误差与投影误差。我们证明这些误差项以特定速率随数据集大小和潜在维度衰减，为缩放定律提供了严格的理论基础。基于理论结果，我们为神经Koopman算子引入了两种轻量级正则化器：协方差损失以帮助稳定学习到的潜在特征，以及逆控制损失以确保模型与物理驱动机制对齐。在六个机器人环境中进行的系统实验结果表明，模型拟合误差遵循所推导的缩放定律，且正则化器提升了动态模型拟合的保真度，同时增强了闭环控制性能。综合而言，我们的研究结果为在学习用于控制的Koopman动力学时，如何在数据收集与模型容量之间分配资源提供了一个简洁的指导方案。