Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models

TL;DR

The study tackles real-time closed-loop control of high-dimensional parametrized PDEs by introducing a nonintrusive DL-ROM framework that couples POD/AE-based dimensionality reduction with a low-dimensional policy $\pi_N$ to map latent states $(y_N)$ and scenario parameters $\boldsymbol{\mu}_s$ to latent controls $(u_N)$. An offline phase generates diverse optimal trajectories via adjoint methods, compresses them into latent coordinates, and trains a joint autoencoder–policy network, while an online phase retrieves control actions through fast forward passes in the latent space; a latent forward model $\varphi_N$ further enables loop closure when online state data are unavailable. The method is validated on two 2D high-dimensional optimal transport problems (vacuum and fluid with obstacles), showing high accuracy, robustness to noise, and dramatic speedups (up to $\sim 3.2\times 10^4$) compared to full-order solvers, illustrating practical impact for real-time sensing-constrained control of distributed systems. The results demonstrate that nonlinear DL-ROMs (DL-ROM, POD+AE) effectively capture transport-dominated dynamics and that latent-loop closures provide reliable control even under sensor failures or delays, highlighting the framework’s potential for sensor-based and multi-scenario applications.

Abstract

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.

Figures (15)

• Figure 1: Feedback control scheme considering multiple optimal control problem resolutions and a high-fidelity full-order model of the dynamical system. Given the current state and the scenario parameters $\boldsymbol{\mu}_s$, the open-loop optimal control and state trajectories are retrieved by solving the optimal control problem. The feedback signal is then recovered by measuring or simulating the state at the next time step.
• Figure 2: Offline phase of the deep learning-based reduced order feedback controller. After generating optimal state and control snapshots through the adjoint method, the state and control autoencoders, namely $\varphi_D^y(\varphi_E^y(\cdot))$ and $\varphi_D^u(\varphi_E^u(\cdot))$, and the surrogate policy $\pi_N$ are trained minimizing the cumulative loss function $J_{\mathrm{NN}} = \lambda_1 J_{\mathrm{rec}}^y + \lambda_2 J_{\mathrm{rec}}^u + J_{\pi_N}$.
• Figure 3: Online phase of the deep learning-based reduced order feedback controller. The optimal full-order control action corresponding to the observed state $\mathbf{y}_h$ in a scenario described by input parameters $\boldsymbol{\mu}_s$ is inferred online through forward passes of $\pi_N$ and the encoding-decoding mappings.
• Figure 4: Online phase of the deep learning-based reduced order feedback controller with latent feedback loop. The optimal full-order control action corresponding to the observed state $\mathbf{y}_h$ in a scenario described by input parameters $\boldsymbol{\mu}_s$ is inferred online through forward-passes of $\pi_N$ and the encoding-decoding mappings. Whenever full-order state data $\mathbf{y}_h$ are not available online, the trained deep learning-based forward model $\varphi_N$ is exploited to predict the state evolution, allowing for a continuous prediction of the control action.
• Figure 5: Test 1.1. Optimal transport in a vacuum. Top left: representation of an optimal state trajectory in a vacuum within the domain $\Omega$, where $\mathbf{y}_0$ stands for the initial density centered at $(\mu_1^0, \mu_2^0) =(-0.45, 0.21)$, while $\mathbf{y}_d(\boldsymbol{\mu}_s)$ represents the target configuration centered at $(\mu_1^d, \mu_2^d) = (0.29, -0.24)$. Top right: discrepancy between current state $\mathbf{y}_h(t)$ and target configuration $\mathbf{y}_d$ centered at $(\mu_1^d, \mu_2^d) = (0.29, -0.24)$ at different time instants in the uncontrolled ($\mathbf{u}_h = \mathbf{0}$) and optimal ($\mathbf{u}_h = \mathbf{u}_h^*$) settings. Other panels: space-varying optimal state and control at $t=0,0.25,0.5,0.75$ related to the scenario parameters $\boldsymbol{\mu}_s = (0.29, -0.24)$. The control velocity fields on $\Omega$ are depicted through vector fields, with the underlying colours corresponding to their magnitude.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4