The optimal control of parameter-dependent partial differential equations plays a central role in many scientific and engineering applications, including wave propagation, structural vibrations, and transport phenomena. Classical numerical approaches rely on high-fidelity discretizations such as finite element or finite difference methods. While these approaches accurately approximate the dynamics of the system, they lead to large-scale dynamical systems whose computational cost becomes prohibitive when repeated evaluations across the parameter space are required, as is typical in optimization and control contexts.
Reduced order models aim to address this challenge by constructing low-dimensional approximations that preserve the essential dynamics of the original system while significantly reducing computational complexity. However, classical projection-based reduced order modeling techniques often exhibit limited performance for transport-dominated phenomena, such as wave equations, due to the slow decay of the Kolmogorov $n$-widths of the associated solution manifold. In such cases, linear approximation spaces require large reduced dimensions to achieve accurate representations.
The proposed research activity focuses on the development of nonlinear reduced order modeling strategies based on machine learning, specifically using autoencoder neural networks. Autoencoders learn low-dimensional latent representations of high-dimensional solution data by encoding the solution manifold into a compact representation and reconstructing the original state from this latent space. This framework enables the approximation of dynamics directly from data and provides a flexible alternative to classical projection-based methods.
In addition to the latent representation obtained through the autoencoder, the project investigates the implementation of a parameter-to-latent map in parameter-dependent optimal control problems as a separate machine learning surrogate that directly associates each parameter value with its corresponding latent representation. This strategy enables rapid online evaluation of parameter-dependent solutions by bypassing the high-dimensional solver entirely.
University of Graz, Leechgasse 34, 8010 Graz, Austria