用神经算子一次性求解最优控制问题，无需迭代或动力学公式。
Optimal Control Operator Perspective and a Neural Adaptive Spectral Method
- 提出控制算子新视角，直接映射输入到最优控制策略。
- 在合成环境和真实数据集上实现显著加速与强泛化能力。
- 适合需快速求解复杂控制问题的研究者与工程师。
最优控制问题（OCP）旨在为动态系统寻找使代价函数最优的控制函数，广泛应用于学术与工业领域。本文提出一种新的实例-解控制算子视角，可一次性求解OCP，不依赖动力学显式表达式或迭代优化过程。控制算子由新型神经算子架构——神经自适应谱方法（NASM）实现，是经典谱方法的推广。理论上，我们给出了NASM对控制算子的逼近误差界。在合成环境和真实世界数据集上的实验验证了该方法的有效性与高效性，包括运行时间的显著加速，以及在分布内和分布外均表现出高质量的泛化能力。
原文摘要 · Abstract (English)
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.
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