Weekly Seminar
Polytopic Autoencoders as a new model order reduction approach with applications in nonlinear controller design
Jan Heiland (TU Ilmenau)
Thu, 25 Jun 2026 • 10:45-11:45h • Pontdriesch 14, room 008 SeMath (host: Michael Herty)

Abstract

The controller design for both nonlinear and large-scale dynamical systems is a challenge and, up to now, no generally applicable and feasible computational approach has been established.

In the first part of the talk, I will comment on the general difficulties and advertise linear-parameter varying (LPV) embeddings as a general basis for development of numerical and theoretical methods. Among others, approximative LPV embeddings allow us to parametrize and reduce the system’s structural complexity (so that controller design becomes easier) while leaving the state space untouched (so that the model expressiveness is preserved).

For such models with a reduced and parametrized nonlinear structure but a high-dimensional state space, several computational approaches are available for controller design. For all of them, however, a low-dimensional parametrization is important. For that we developed the concept of polytopic autoencoders that reliably outperform standard model order reduction like POD (Proper Orthogonal Decomposition) at very low dimensions and that provide additional beneficial structures in the LPV parametrization. The main idea is that reconstruction happens in a polytope rather than in a linear space and the realization is done in specially developed neural network architectures.

In a second part, I will introduce the concept of polytopic autoencoders and highlight analytical properties of the employed neural network architecture.

Finally, I will present the applications in nonlinear controller design and try to make connections to other fields such as optimization and modeling where affine parameter dependencies are considered.

References:

  1. Heiland, Jan / Kim, Yongho: Polytopic autoencoders with smooth clustering for reduced-order modeling of flows (2025) https://doi.org/10.1016/j.jcp.2024.113526 — https://arxiv.org/abs/2403.18044

  2. Heiland, Jan / Kim, Yongho / Werner, Steffen W. R.: Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback controller design (2025) https://doi.org/10.1007/s10444-025-10269-1 — https://arxiv.org/abs/2401.10620