Optimal Control Operator Perspective and a Neural Adaptive Spectral Method

TL;DR

This paper introduces a novel neural spectral method that provides a one-shot solution to optimal control problems, bypassing traditional iterative methods and demonstrating high efficiency and generalization in experiments.

Contribution

The paper proposes a new control operator perspective and a neural adaptive spectral method (NASM) that efficiently solves OCPs without explicit dynamics or iterative optimization.

Findings

Significant speedup in computation time.

High-quality generalization to out-of-distribution data.

Theoretical validation with approximation error bounds.

Abstract

Optimal control problems (OCPs) involve finding a control function for a dynamical system such that a cost functional is optimized. It is central to physical systems in both academia and industry. In this paper, we propose a novel instance-solution control operator perspective, which solves OCPs in a one-shot manner without direct dependence on the explicit expression of dynamics or iterative optimization processes. The control operator is implemented by a new neural operator architecture named Neural Adaptive Spectral Method (NASM), a generalization of classical spectral methods. We theoretically validate the perspective and architecture by presenting the approximation error bounds of NASM for the control operator. Experiments on synthetic environments and a real-world dataset verify the effectiveness and efficiency of our approach, including substantial speedup in running time, and high-quality in- and out-of-distribution generalization.