Overcoming slow Kolmogorov width decay in parametric optimal control via neural network surrogates

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.