ANOVA approximation for high-dimensional control systems
Abstract
We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz--Zygmund inequality for the resulting ansatz space, we prove a probabilistic L2-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.
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BibTeX
@misc{kircheis2026anova,
title = {{ANOVA approximation for high-dimensional control systems}},
author = {Melanie Kircheis and Daniel Potts and Manuel Schaller and Karl Worthmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19858},
eprint = {2610.19858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19858},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - ANOVA approximation for high-dimensional control systems
AU - Melanie Kircheis
AU - Daniel Potts
AU - Manuel Schaller
AU - Karl Worthmann
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19858
UR - https://arxiv.org/abs/2610.19858
AB - We study the data-driven approximation of control-affine systems, where the drift and each control vector field exhibit a low-dimensional structure. While the ANOVA (ANalysis Of VAriance) decomposition is tailored to low-dimensional structures, a straightforward implementation does not exploit control affinity, rendering the approximation computationally expensive. We propose a coupled least squares approach for given state-control samples, whose ansatz is affine linear in the control by construction and can be realized by a preprocessing of the ANOVA index sets. Based on a Marcinkiewicz--Zygmund inequality for the resulting ansatz space, we prove a probabilistic L_2-error bound with explicitly given constants. Separate error control on drift and input vector fields renders the approach flexible to, e.\,g., parameter changes. Finally, we numerically illustrate the identification of the coupling structure via global sensitivity indices on discretizations of a nonlinear advection-diffusion equation, the shallow water equations and the viscous Burgers equation.
ER -