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

We address data assimilation for linear and nonlinear dynamical systems via the so-called model reference adaptive system.

Continuing our theoretical developments, 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.