A Data-Assimilation-Augmented Optimization Framework for Parameter Estimation in Dynamical Systems

Abstract

Parameter estimation in nonlinear dynamical systems from observational data is a fundamental inverse problem with applications in many disciplines. In practice, this is further complicated by the fact that observations are often noisy, sparse, and available only for a subset of the state variables. Furthermore, the initial condition (IC) may be unknown or inaccurate, causing further complications for chaotic systems with sensitive dependence on initial conditions. In this work, we develop a data-assimilation-augmented optimization framework for parameter estimation in ordinary differential equations using partial state observations. The method introduces a nudged system driven by the available observed component and estimates the unknown parameters by minimizing a cost functional, defined as a time-delayed mismatch between the observations and the corresponding observed component of the nudged solution over the admissible parameter space. Since the nudged system can be arbitrarily initialized, this approach eliminates the dependence on accurate IC. Using the Lorenz-63 system as a test case, we establish theoretical results showing synchronization of the nudged solution under parameter agreement, stability under parameter mismatch, and well-posedness of the data-to-parameter inverse map under suitable nondegeneracy conditions. Structural & practical identifiability, and Sobol sensitivity analyses are incorporated to assess which parameters can be reliably estimated from the observations. Numerical experiments in both chaotic and non-chaotic regimes show that this framework accurately recovers parameters from noisy partial observations. Comparisons with an on-the-fly parameter learning method and with Bayesian MCMC estimation demonstrate that the proposed method remains accurate under partial observations and higher noise levels while requiring substantially lower computational cost.