We address data assimilation for linear and nonlinear dynamical systems via the so-called \emph{model reference adaptive system}. Continuing our theoretical developments in \cite{Tram_Kaltenbacher_2021}, we deliver the first practical implementation of this approach for online parameter identification with time series data. Our semi-implicit scheme couples a modified state equation with a parameter evolution law that is driven by model-data residuals. We demonstrate four benchmark problems of increasing complexity: the Darcy flow, the Fisher-KPP equation, a nonlinear potential equation and finally, an Allen-Cahn type equation. Across all cases, explicit model reference adaptive system construction, verified assumptions and numerically stable reconstructions underline our proposed method as a reliable, versatile tool for data assimilation and real-time inversion.
@article{arxiv.2602.10920,
title = {Data assimilation via model reference adaptation for linear and nonlinear dynamical systems},
author = {Benedikt Kaltenbach and Christian Aarset and Tram Thi Ngoc Nguyen},
journal= {arXiv preprint arXiv:2602.10920},
year = {2026}
}