Self-Supervised Amortized Neural Operators for Optimal Control: Scaling Laws and Applications

TL;DR

This paper introduces a self-supervised neural operator approach for real-time optimal control, leveraging scaling laws and applications to improve efficiency in low-intrinsic-dimension problems while analyzing limitations in complex scenarios.

Contribution

It presents a novel neural operator framework for open-loop and closed-loop optimal control, integrating with MPC and deriving scaling laws for generalization error.

Findings

Efficient real-time control in low-intrinsic-dimension regimes.

Accuracy degrades with increasing problem complexity.

Theoretical scaling laws relate error to problem complexity.

Abstract

Optimal control provides a principled framework for transforming dynamical system models into intelligent decision-making, yet classical computational approaches are often too expensive for real-time deployment in dynamic or uncertain environments. In this work, we propose a method based on self-supervised neural operators for open-loop optimal control problems. It offers a new paradigm by directly approximating the mapping from system conditions to optimal control strategies, enabling instantaneous inference across diverse scenarios once trained. We further extend this framework to more complex settings, including dynamic or partially observed environments, by integrating the learned solution operator with Model Predictive Control (MPC). This yields a solution-operator learning method for closed-loop control, in which the learned operator supplies rapid predictions that replace the potentially time-consuming optimization step in conventional MPC. This acceleration comes with a quantifiable price to pay. Theoretically, we derive scaling laws that relate generalization error and sample/model complexity to the intrinsic dimension of the problem and the regularity of the optimal control function. Numerically, case studies show efficient, accurate real-time performance in low-intrinsic-dimension regimes, while accuracy degrades as problem complexity increases. Together, these results provide a balanced perspective: neural operators are a powerful novel tool for high-performance control when hidden low-dimensional structure can be exploited, yet they remain fundamentally constrained by the intrinsic dimensional complexity in more challenging settings.