Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems

2025/07/01 by Christian Engwer, Engwer, Christian, Mario Ohlberger +3
Physics and Astronomy · Engineering · Computer Science · #Model Reduction and Neural Networks #Control and Stability of Dynamical Systems #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2507.00678

Abstract

We investigate parametrized variational problems where for each parameter the solution may originate from a different parameter-dependent function space. Our main motivation is the theory of Friedrichs' systems, a large abstract class of linear PDE-problems whose solutions are sought in operator- (and thus parameter-)dependent graph spaces. Other applications include function spaces on parametrized domains or discretizations involving data-dependent stabilizers. Concerning the set of all parameter-dependent solutions, we argue that in these cases the interpretation as a "solution manifold" widely adopted in the model order reduction community is no longer applicable. Instead, we propose a novel framework based on the theory of fiber bundles and explain how established concepts such as approximability generalize by introducing a Sectional Kolmogorov N-width. Further, we prove exponential approximation rates of this N-width if a norm equivalence criterion is fulfilled. Applying this result to problems with Friedrichs' structure then gives a sufficient criterion that can be easily verified.

Citations

• Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems
• Performance bounds for Reduced Order Models with Application to Parametric Transport
• Parametric model order reduction for a wildland fire model via the shifted POD based deep learning method
• Bridging Traditional and Machine Learning-based Algorithms for Solving PDEs: The Random Feature Method
• Nonlinear Model Order Reduction using Diffeomorphic Transformations of a\n Space-Time Domain
• Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking
• A registration method for model order reduction: data compression and\n geometry reduction
• Free Form Deformation, mesh morphing and reduced order methods: enablers for efficient aerodynamic shape optimization

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