Parameter-dependent optimal control problems arise when the dynamics, target states, physical coefficients, or other components of a controlled system vary with a set of parameters. They are particularly important in applications where an optimal control problem must be solved repeatedly for many different configurations. Although high-fidelity numerical methods can provide accurate solutions, repeatedly solving the underlying partial differential equations and optimality systems can quickly become computationally prohibitive. This makes the construction of efficient surrogate models a central question for many-query optimal control problems.
A natural strategy is to use reduced-order models, replacing the original high-dimensional problem by an approximation in a much smaller space. An example for such a method is discussed in this blogpost. However, this approach faces a limitation when the family of solutions cannot be represented accurately by low-dimensional linear spaces. This phenomenon can be quantified through the Kolmogorov $N$-width, which measures how well a solution manifold can be approximated by an $N$-dimensional linear space. When the Kolmogorov width decays slowly, increasing the dimension of a classical reduced basis leads only to a gradual improvement in accuracy and can considerably reduce the computational advantages of model reduction.
This difficulty was the motivation for the Short-Term Scientific Mission (STSM) within COST Action CA24136 InterCoML, giving as result the work "Overcoming slow Kolmogorov width decay in parametric optimal control via neural network surrogates" , by Hendrik Kleikamp, Martin Lazar, and Juan Ricardo Muñoz, that is nowadays under revision.
The work considers parameter-dependent linear-quadratic optimal control problems for which the optimal solution can be characterized through the final-time adjoint state. This observation provides a particularly useful perspective: rather than approximating the entire state and control trajectories directly, the problem can be reduced to approximating the parameter-dependent final-time adjoint, from which the optimal state and control can subsequently be recovered.
The theoretical analysis identifies structural obstructions to classical linear reduced-order modeling. For distributed control of the heat equation, we show that a slowly decaying Kolmogorov width of the parameter-dependent target manifold can be inherited by the manifold of final-time adjoints.
In particular, moving and discontinuous targets generate a transport-dominated structure that remains difficult to approximate with low-dimensional linear spaces, despite the regularizing properties of the heat equation. Thus, the difficulty is not simply caused by the numerical discretization or by the dynamics: it is an intrinsic feature of the parameter dependence itself.
Motivated by this analysis, we investigate nonlinear machine-learning surrogates as an alternative. A U-Net architecture is used to learn the map from parameter-dependent spatial fields (target states and diffusivity fields) to the corresponding optimal final-time adjoint. The architecture is shown in Figure 1.
Figure 1: U-Net architecture with parametric fields as inputs and optimal final time adjoints as output.
This formulation allows the network to retain information about the position and geometry of localized structures instead of requiring this information to be reconstructed from a low-dimensional parameter vector. Moreover, since the network predicts the final-time adjoint itself, a residual-based a posteriori error estimator remains available to assess the quality of the approximation.
The numerical experiments support this approach. Across two two-dimensional optimal control problems, including a more challenging example combining a moving target, a discontinuous parameter-dependent diffusivity, and a localized control region, the U-Net consistently provides the most accurate approximation among the considered linear and nonlinear methods.
Particularly relevant for practical applications is its performance in the small-data regime. In the reported experiments, a U-Net trained with only 100 snapshots already achieves average final-time-adjoint errors more than one order of magnitude smaller than competing approaches trained with 1000 snapshots. Once trained, prediction of the final-time adjoint requires less than one millisecond, while recovery of the state and control still provides substantial computational speedups over the full-order problem. Quantitative results of the U-Net are shown in Figure 2.
Figure 2: Results of the U-Net surrogate for three test parameters. Columns from left to right: diffusivity, target state, full-order model final time adjoint, U-Net prediction, and pointwise error. The orange square depicts the control domain.
An important part of the development of this research was supported by a Short-Term Scientific Mission (STSM) within COST Action CA24136 InterCoML -- Control and Machine Learning. The STSM brought Juan Ricardo Muñoz to the University of Graz to work with Hendrik Kleikamp on nonlinear model-reduction strategies for parameter-dependent optimal control.
The original research plan focused particularly on autoencoders and parameter-to-latent mappings as alternatives to classical linear reduced-order models in situations characterized by slowly decaying Kolmogorov widths. The collaboration made possible by the STSM helped accelerate the research beyond its initial formulation. Working in the same place allowed numerical experiments to be developed and discussed rapidly, limitations of the initial approaches to be identified, and new ideas to be tested directly.
In particular, the investigation of autoencoder and parameter-to-latent strategies contributed to a broader comparison of nonlinear representations and ultimately to the field-based U-Net approach developed in the paper. The resulting work therefore illustrates one of the main benefits of STSMs: they provide researchers with dedicated time and a shared environment in which preliminary ideas can be transformed into concrete research directions and collaborative results.
Additionally, the environment of the IDea_Lab at the University of Graz, where Hendrik Kleikamp is based, provided a valuable setting for exchanging ideas at the interface of mathematical modeling, numerical analysis, reduced-order modeling, and machine learning.
The STSM thus contributed not only to accelerating a specific research project, but also to strengthening connections between researchers and institutions within the InterCoML network. Such opportunities illustrate the broader role of COST Actions in creating research networks in which short scientific visits can initiate new discussions, accelerate ongoing projects, and create connections that extend beyond the duration of the visit itself.
University of Graz, Leechgasse 34, 8010 Graz, Austria
Sveuciliste U Dubrovniku, Branitelja Dubrovnika 41, 20000 Dubrovnik, Croatia
