Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.
We propose a continuous-time algorithm for zeroth-order global optimization of smooth functions on compact manifolds that synthesizes Riemannian gradient dynamics using only objective measurements, without evaluating gradients. The algorithm is formulated as an ordinary differential equation driven by piecewise-smooth...
Mahmoud Abdelgalil, Miroslav Krstić, J. Poveda· 1 citation
The Kolmogorov
n
-width is an established benchmark to judge the approximation properties of the reduced linear spaces arising from reduced order models (ROMs). Although immensely successful in the elliptic regime, this width shows unsatisfactory slow convergence rates for transport dominated problems. While this...
This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of s...
A coupled perturbation analysis for the entire PPMD procedure identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.
Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approac...
J. A. Hernández, S. A. de Parga, R. Rossi· 1 citation
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