Sequential Bayesian parameter-state estimation in dynamical systems with noisy and incomplete observations via a variational framework

Online joint estimation of unknown parameters and states in a dynamical system with uncertainty quantification is crucial in many applications. For example, digital twins dynamically update their knowledge of model parameters and states to support prediction and decision-making. Reliability and computational speed are vital for DTs. Online parameter-state estimation ensures computational efficiency, while uncertainty quantification is essential for making reliable predictions and decisions. In parameter-state estimation, the joint distribution of the state and model parameters conditioned on the data, termed the joint posterior, provides accurate uncertainty quantification. Because the joint posterior is generally intractable to compute, this paper presents an online variational inference framework to compute its approximation at each time step. The approximation is factorized into a marginal distribution over the model parameters and a state distribution conditioned on the parameters. This factorization enables recursive updates through a two-stage procedure: first, the parameter posterior is approximated via variational inference; second, the state distribution conditioned on the parameters is computed using Gaussian filtering based on the estimated parameter posterior. The algorithmic design is supported by a theorem establishing upper bounds on the joint posterior approximation error. Numerical experiments demonstrate that the proposed method (i) matches the performance of the joint particle filter in low-dimensional problems, accurately inferring both unobserved states and unknown parameters of dynamical and observation models; (ii) remains robust under noisy, partial observations and model discrepancies in a chaotic Lorenz 96 system; and (iii) scales effectively to a high-dimensional convection-diffusion system, where it outperforms the joint ensemble Kalman filter.