From reduced bases to U-Nets: Machine Learning for parametric optimal control
In this talk, we consider optimal control problems whose dynamics and objective functional depend on parameters. Solving such problems for many parameter values is often computationally prohibitive. To address this challenge, we consider reduced-order models (ROMs) that accelerate computations while retaining rigorous accuracy guarantees, including a posteriori error estimates.
The projection-based reduced basis ROMs considered here can be further accelerated through the incorporation of machine learning techniques. By combining classical model reduction with machine learning, the a posteriori error estimator can be transferred directly to the machine learning predictions. Moreover, the full-order model, reduced basis ROM, and machine learning surrogate can be organized into an adaptive model hierarchy, in which the different models are trained and employed in an adaptive manner.
In the final part of the talk, we present recent work on the use of U-Nets for nonlinear surrogate modeling in problems characterized by slowly decaying Kolmogorov widths, where linear reduced models are known to struggle. To assess their performance, we compare U-Nets with alternative approaches, including autoencoder architectures and parametric decoders, on challenging optimal control problems.