通过神经网络代理克服参数最优控制中缓慢的柯尔莫哥洛夫宽度衰减
Overcoming slow Kolmogorov width decay in parametric optimal control via neural network surrogates
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中文总结 AI 辅助
研究参数化线性二次最优控制问题，针对传统线性降阶模型不足，提出基于U-Net的非线性代理，通过后验误差估计器验证结果，经实验表明该代理精度高且所需训练样本少。
中文摘要 AI 辅助
本文研究参数化线性二次最优控制问题，其解可由最优终端时间伴随状态唯一表征。以热方程的分布式控制为例，理论结果表明，若参数依赖的目标状态存在缓慢的柯尔莫哥洛夫宽度衰减，那么参数空间上终端时间伴随状态的流形也会如此。传统线性降阶模型需大的降维空间以保证精度，效率低。为克服线性模型局限，讨论基于U-Net的非线性代理，将参数场映射到近似终端时间伴随状态。合适的后验误差估计器适用于U-Net近似，可验证代理结果。通过两个数值实验展示U-Net代理潜力，并与文献中的线性和非线性方法比较。结果表明，U-Net在考虑的方法中始终具有最高精度，且所需训练样本显著更少。
英文摘要
In this paper we deal with parametric, linear-quadratic optimal control problems in which the solution can be uniquely characterized by the optimal final time adjoint state. As a motivating example, we establish theoretical results showing that for distributed control of the heat equation, the manifold of final time adjoints over the parameter space exhibits a slow decay of its Kolmogorov width if this was already the case for the parameter-dependent target states. Traditional linear reduced-order models would thus require a large reduced space in order to guarantee a sufficient accuracy, making them inefficient in this application. To overcome the limitation of linear models, we discuss a nonlinear surrogate based on U-Nets that maps parametric fields to approximate final time adjoints. We show that a suitable a posteriori error estimator remains applicable to the U-Net approximation and can be used to certify the surrogate results. Through two extensive numerical experiments, we show the potential of the U-Net surrogate and compare it with several linear and nonlinear methods from the literature. The results show that the U-Net consistently achieves the highest accuracy among the methods considered while requiring significantly fewer training samples.
