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Data assimilation via model reference adaptation for linear and nonlinear dynamical systems

Benedikt Kaltenbach, Christian Aarset, Tram Thi Ngoc Nguyen

TL;DR

This work develops and implements a model reference adaptive system (MRAS) for online parameter identification in linear and nonlinear parabolic PDEs, advancing beyond prior linear-parameter models to nonlinear-in-state and nonlinear-in-parameter scenarios. The authors couple a semi-implicit state evolution with an online parameter update driven by model-data residuals, proving convergence under structural assumptions and presenting an explicit discretization via continuous Galerkin FEM and semi-implicit time stepping. Four benchmark problems (Darcy flow, Fisher–KPP, nonlinear potential, and modified Allen–Cahn) demonstrate the method’s ability to recover spatially varying parameters and reconstruct states in real time, even under measurement noise. The results show robust parameter convergence and accurate state estimates, with practical implications for real-time inversion and data assimilation in complex, nonlinear PDE systems. The work also outlines future extensions to alternative observation operators and higher-order time-stepping schemes to broaden MRAS applicability.

Abstract

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.

Data assimilation via model reference adaptation for linear and nonlinear dynamical systems

TL;DR

This work develops and implements a model reference adaptive system (MRAS) for online parameter identification in linear and nonlinear parabolic PDEs, advancing beyond prior linear-parameter models to nonlinear-in-state and nonlinear-in-parameter scenarios. The authors couple a semi-implicit state evolution with an online parameter update driven by model-data residuals, proving convergence under structural assumptions and presenting an explicit discretization via continuous Galerkin FEM and semi-implicit time stepping. Four benchmark problems (Darcy flow, Fisher–KPP, nonlinear potential, and modified Allen–Cahn) demonstrate the method’s ability to recover spatially varying parameters and reconstruct states in real time, even under measurement noise. The results show robust parameter convergence and accurate state estimates, with practical implications for real-time inversion and data assimilation in complex, nonlinear PDE systems. The work also outlines future extensions to alternative observation operators and higher-order time-stepping schemes to broaden MRAS applicability.

Abstract

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.
Paper Structure (35 sections, 9 theorems, 43 equations, 11 figures, 5 tables)

This paper contains 35 sections, 9 theorems, 43 equations, 11 figures, 5 tables.

Table of Contents

  1. Abstract
  2. Introduction
  3. Outline
  4. Model reference adaptive system for data assimilation
  5. Underlying parametric equation and time series data
  6. Data assimilation via model reference adaptive system
  7. Asymptotic convergence of the MRAS
  8. MRAS discretisation and update rule
  9. Finite-element discretisation
  10. Semi-implicit Euler time stepping
  11. Update and preconditioner
  12. Darcy flow: the linear $a$-problem
  13. MRAS analysis
  14. Numerical results
  15. Implementation setup
  16. ...and 20 more sections

Key Result

Proposition 1

Let Assumption ass-1 be fulfilled. Then the following statements on the reconstructed parameter $q$, the state $u$ and the corresponding errors $\varepsilon_q:=q-q^\dagger$, $\varepsilon_u:=u-u^\dagger$ hold true:

Figures (11)

  • Figure 1: Parameters in benchmark examples, including Darcy flow, Fisher-KPP, nonlinear potential and modified Allen-Cahn equation.
  • Figure 2: Schematic of data assimilation for the state $u$ over assimilation windows $t_{i+1}-t_i$. MRAS iteratively reconstructs the unknown parameters driven by the assimilation of the state.
  • Figure 3: MRAS workflow for dynamic update laws
  • Figure 4: Darcy flow. Visualization of evolution of the state $u$ and diffusion parameter $a$ computed from MRAS (top) and exact quantities $u^\dagger, a^\dagger$ (bottom).
  • Figure 5: Fisher-KPP. Visualization of evolution of the state $u$ and diffusion parameter $a$ ran with clean data (left) and 3% noisy data (right).
  • ...and 6 more figures

Theorems & Definitions (20)

  • Remark 1
  • Proposition 1: Convergence
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • proof
  • Proposition 3
  • proof
  • Corollary 2
  • ...and 10 more