Scaling Law of Neural Koopman Operators

TL;DR

This paper establishes and empirically validates scaling laws for neural Koopman operators, linking sample size, latent dimension, and control quality, and introduces regularizers to improve model performance in robotic control tasks.

Contribution

The paper derives theoretical scaling laws for neural Koopman operators and introduces regularizers that enhance model stability and control accuracy, providing practical guidance for data and model capacity allocation.

Findings

Model fitting error follows derived scaling laws.

Regularizers improve dynamic model fidelity.

Enhanced closed-loop control performance in robotic environments.

Abstract

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.