Tobias Breiten

Tobias Breiten, Christopher Beattie, Serkan Gugercin

This paper develops an interpolatory framework for weighted-$\mathcal{H}_2$ model reduction of MIMO dynamical systems. A new representation of the weighted-$\mathcal{H}_2$ inner products in MIMO settings is introduced and used to derive associated first-order necessary conditions satisfied by optimal weighted-$\mathcal{H}_2$ reduced-order models. Equivalence of these new interpolatory conditions with earlier Riccati-based conditions given by Halevi is also shown. An examination of realizations for equivalent weighted-$\mathcal{H}_2$ systems leads then to an algorithm that remains tractable for large state-space dimension. Several numerical examples illustrate the effectiveness of this approach and its competitiveness with Frequency Weighted Balanced Truncation and an earlier interpolatory approach, the Weighted Iterative Rational Krylov Algorithm.

Cankat Tilki, Tobias Breiten, Serkan Gugercin

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.

Edmund Ross, Claudia Drygala, Leonhard Schwarz et al.

In this work, we explore the use of compact latent representations with learned time dynamics ('World Models') to simulate physical systems. Drawing on concepts from control theory, we propose a theoretical framework that explains why projecting time slices into a low-dimensional space and then concatenating to form a history ('Tokenization') is so effective at learning physics datasets, and characterise when exactly the underlying dynamics admit a reconstruction mapping from the history of previous tokenized frames to the next. To validate these claims, we develop a sequence of models with increasing complexity, starting with least-squares regression and progressing through simple linear layers, shallow adversarial learners, and ultimately full-scale generative adversarial networks (GANs). We evaluate these models on a variety of datasets, including modified forms of the heat and wave equations, the chaotic regime 2D Kuramoto-Sivashinsky equation, and a challenging computational fluid dynamics (CFD) dataset of a 2D Kármán vortex street around a fixed cylinder, where our model is successfully able to recreate the flow.