Continuous monitoring and real-time control of high-dimensional distributed systems are often crucial in applications to ensure a desired physical behavior, without degrading stability and system performances. Traditional feedback control design that relies on full-order models, such as high-dimensional state-space representations or partial differential equations, fails to meet these requirements due to the delay in the control computation, which requires multiple expensive simulations of the physical system. The computational bottleneck is even more severe when considering parametrized systems, as new strategies have to be determined for every new scenario. To address these challenges, we propose a real-time closed-loop control strategy enhanced by nonlinear non-intrusive Deep Learning-based Reduced Order Models (DL-ROMs). Specifically, in the offline phase, (i) full-order state-control pairs are generated for different scenarios through the adjoint method, (ii) the essential features relevant for control design are extracted from the snapshots through a combination of Proper Orthogonal Decomposition (POD) and deep autoencoders, and (iii) the low-dimensional policy bridging latent control and state spaces is approximated with a feedforward neural network. After data generation and neural networks training, the optimal control actions are retrieved in real-time for any observed state and scenario. In addition, the dynamics may be approximated through a cheap surrogate model in order to close the loop at the latent level, thus continuously controlling the system in real-time even when full-order state measurements are missing. The effectiveness of the proposed method, in terms of computational speed, accuracy, and robustness against noisy data, is finally assessed on two different high-dimensional optimal transport problems, one of which also involving an underlying fluid flow.

翻译：高维分布式系统的持续监测与实时控制在应用中至关重要，以确保期望的物理行为，同时不损害系统稳定性与性能。传统依赖于全阶模型（如高维状态空间表示或偏微分方程）的反馈控制设计，由于控制计算延迟（需对物理系统进行多次昂贵的仿真）而无法满足这些要求。当考虑参数化系统时，计算瓶颈更为严重，因为每个新场景都需要重新制定控制策略。为应对这些挑战，我们提出一种通过非线性非侵入式深度学习降阶模型增强的实时闭环控制策略。具体而言，在离线阶段：（i）通过伴随方法为不同场景生成全阶状态-控制对；（ii）通过本征正交分解与深度自编码器的组合，从快照数据中提取控制设计所需的关键特征；（iii）使用前馈神经网络近似连接隐式控制空间与状态空间的低维策略。在数据生成与神经网络训练完成后，可针对任意观测状态与场景实时获取最优控制动作。此外，系统动力学可通过廉价代理模型进行近似，从而在隐式层面实现闭环控制，即使在全阶状态测量缺失时仍能持续实时控制系统。最后，通过两个不同的高维最优输运问题（其中一个还涉及底层流体流动），从计算速度、精度及对噪声数据的鲁棒性等方面评估了所提方法的有效性。