Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models

TL;DR

The paper tackles the challenge of real-time optimal control for high-dimensional parametrized PDEs by introducing a non-intrusive DL-ROM framework. It combines dimensionality reduction (POD and autoencoders) with neural networks to map scenario parameters to latent reduced states and controls, followed by decoding to full-order quantities in an offline-online workflow. Across steady and time-dependent Navier–Stokes flows and an active thermal cooling problem, the method achieves sub-4% accuracy with online times in the order of 0.001–0.01 s, providing dramatic speedups over full-order solvers and enabling rapid evaluation for many scenarios. This approach offers a versatile, scalable solution for multi-scenario PDE-constrained OCPs and opens avenues for extensions to sensor data, robust control, and data-driven surrogates.

Abstract

Steering a system towards a desired target in a very short amount of time is challenging from a computational standpoint. Indeed, the intrinsically iterative nature of optimal control problems requires multiple simulations of the physical system to be controlled. Moreover, the control action needs to be updated whenever the underlying scenario undergoes variations. Full-order models based on, e.g., the Finite Element Method, do not meet these requirements due to the computational burden they usually entail. On the other hand, conventional reduced order modeling techniques such as the Reduced Basis method, are intrusive, rely on a linear superimposition of modes, and lack of efficiency when addressing nonlinear time-dependent dynamics. In this work, we propose a non-intrusive Deep Learning-based Reduced Order Modeling (DL-ROM) technique for the rapid control of systems described in terms of parametrized PDEs in multiple scenarios. In particular, optimal full-order snapshots are generated and properly reduced by either Proper Orthogonal Decomposition or deep autoencoders (or a combination thereof) while feedforward neural networks are exploited to learn the map from scenario parameters to reduced optimal solutions. Nonlinear dimensionality reduction therefore allows us to consider state variables and control actions that are both low-dimensional and distributed. After (i) data generation, (ii) dimensionality reduction, and (iii) neural networks training in the offline phase, optimal control strategies can be rapidly retrieved in an online phase for any scenario of interest. The computational speedup and the high accuracy obtained with the proposed approach are assessed on different PDE-constrained optimization problems, ranging from the minimization of energy dissipation in incompressible flows modelled through Navier-Stokes equations to the thermal active cooling in heat transfer.

Figures (22)

• Figure 1: Test 1.1. Indirect approach for a PDE-constrained optimization problem in the case of steady fluid flow control. Here $\mathbf{u}_h^{(k)}$ is the suboptimal boundary control at the $k$-th iteration of the optimization loop; $\mathbf{y}_h^{(k)}$ denotes the corresponding flow velocity computed by solving the steady Navier-Stokes equations; $\mathbf{p}_h^{(k)}$ is the corresponding adjoint variable, resulting from the adjoint equation; $\mathbf{u}_h^{(k+1)}$ is the boundary control updated through the steepest descent method with step size $\eta$.
• Figure 2: Test 1.1. Offline-online decomposition required to solve PDE-constrained optimization problems faster exploiting a reduced basis solver, in the case of steady flow control. Offline: generation of optimal state, adjoint and control snapshots for random scenarios and reduction of the first-order optimality conditions in the KKT system. Online: resolution of the reduced KKT system to compute the optimal state, adjoint and control for a new scenario of interest. In general, the reduced KKT system is solved through the Newton method with reduced Jacobian matrix $J_{\mathbf{F}_N}$ and right-hand side $\mathbf{F}_N$ properly approximated.
• Figure 3: Test 1.2. Deep learning-based reduced order model architecture to solve parametrized optimal control problems in real-time. Optimal state and control snapshots are generated and reduced through a combination of POD and autoencoders. Moreover, the map $\varphi$ from time and scenario parameters to reduced optimal pair is modeled through a feed-forward neural network. After properly training the networks in the offline phase, the full-order optimal pair corresponding to a new time instant $t^{\text{new}}$ and a new scenario $\boldsymbol{\mu}_s^{\text{new}}$ is retrieved online by a forward pass through $\varphi$ and the decoders.
• Figure 4: Test 1. Flow control. Mesh exploited to generate high-fidelity snapshots both in steady and time-dependent flow control test cases, along with the boundaries considered in Navier-Stokes equations. In particular, $\Gamma_{\mathrm{in}}$ in red is the inflow boundary where the fluid enters the channel, $\Gamma_{\mathrm{out}}$ in blue is the outflow, $\Gamma_{\mathrm{walls}}$ in black are the walls bounding the channel from the top and the bottom, $\Gamma_{\mathrm{obs}}$ in green and the control region $\Gamma_c$ in magenta are the two portions of the obstacle boundary.
• Figure 5: Test 1.1. Steady flow control. Optimal control, velocity and pressure obtained for $\left\lVert\mathbf{v}_{\mathrm{in}}\right\rVert = 31.16 m s^{-1}$ and $\alpha_{\mathrm{in}} = -0.53$ radians through the high-fidelity OCP solver. The velocity on $\Omega$ is depicted through a scalar field with colours corresponding to its norm, while the control on $\Gamma_c$ is represented through a vector field.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3