Multi-level Optimal Control with Neural Surrogate Models

TL;DR

This work addresses the computational burden of jointly optimizing actuator design and control for PDE-governed systems by introducing neural network surrogates for the parametric, lower-level value function $V(z_0,r)$ governed by a parameter-dependent ARE. It develops both unstructured and structured surrogates, with the latter enforcing a PSD structure via a Cholesky factor $L_{\theta}(r)$ so that $\Pi(r) \approx L_{\theta}(r)L_{\theta}^\top(r)$, and demonstrates improved accuracy and positivity preservation. The outer optimization uses surrogate-based max-min solvers, including projected gradient descent-ascent and a consensus-based saddle-point method, validated on an actuator-location problem for heat control where the surrogate-enabled methods converge to similar, physically meaningful actuator placements. The approach significantly accelerates evaluations and enables scalable optimization, with potential applicability to nonlinear distributed-parameter systems in future work.

Abstract

Optimal actuator and control design is studied as a multi-level optimisation problem, where the actuator design is evaluated based on the performance of the associated optimal closed loop. The evaluation of the optimal closed loop for a given actuator realisation is a computationally demanding task, for which the use of a neural network surrogate is proposed. The use of neural network surrogates to replace the lower level of the optimisation hierarchy enables the use of fast gradient-based and gradient-free consensus-based optimisation methods to determine the optimal actuator design. The effectiveness of the proposed surrogate models and optimisation methods is assessed in a test related to optimal actuator location for heat control.

Figures (5)

• Figure 1: True $\max_{\|z_{0}\| = 1}V(z_{0}, r)$, structured and unstructured surrogates $\max_{\|z_{0}\| = 1}V_{\theta}(z_{0}, r)$ for $m=1$ and $n = 3$. Both surrogates effectively approximate the value function without violating the non-negativeness constraint, the structured surrogate generates a more accurate solution.
• Figure 2: The heat map of absolute training error using the structured NN for $n =10$ and $m = 2$, with the surrogate constructed as indicated in Subsec. \ref{['subsec:NNPiMatrix']}.
• Figure 3: The heat map corresponding to the real worst-case value function.
• Figure 4: The heat map corresponding to the NN surrogate worst-case value function. Red: solution trajectories of PGDA, Magenta: consensus point trajectories of CBO-SP. Initial points are depicted with a circle and end points with a star.
• Figure 5: Closed-loop performance of the optimised actuators at $r_{CBO-SP} = [2.0472, 1.1830]$ and non-optimised ones placed at $r = [0, 0]^{\top}$. The first three modes are demonstrated with circle, triangle and cross, respectively.