Optimal Control Operator Perspective and a Neural Adaptive Spectral Method
This addresses optimal control for dynamical systems in academia and industry, offering a potentially more efficient approach, though it appears incremental as it generalizes classical spectral methods.
The paper tackles optimal control problems by proposing a novel instance-solution control operator perspective that solves them in a one-shot manner without iterative optimization, and implements it with a Neural Adaptive Spectral Method, achieving substantial speedup and high-quality generalization in experiments.
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.