Parameter Estimation for Statistical Finite Elements
Abstract. When using differential equations to model a dynamical system, uncertainty arises from data limitations, parameter selection, and mismatch between the model and real-world dynamics. To address these challenges. We extend the Statistical Finite Element Method (StatFEM) to incorporate physics-based model parameter uncertainty. We propose a nested Bayesian filtering scheme: an outer Unscented Kalman Filter (UKF) propagates a Gaussian approximation of the parameter posterior, where representative points condition state-level StatFEM filters. The marginal likelihood from each state-level filter is then used to perform the Bayesian update for the parameter distribution in the UKF. This nested Gaussian structure results in an efficient, online algorithm for the joint estimation of state, physical parameters, and structural uncertainty. We demonstrate our method on two time-dependent problems. First, the linear heat equation is a pedagogical example, where we also compare to both traditional Particle Markov Chain Monte Carlo and Sequential Monte Carlo Squared. Second is the nonlinear Korteweg–de Vries equation. In the linear case, we reveal a trade-off between statistical optimality and physical interpretability—particle methods explore the full misspecified posterior, but a fast, recursive Gaussian approximation efficiently recovers the true physical parameters. Finally, our nonlinear application demonstrates its capability to handle complex dynamics where Monte Carlo schemes may be computationally prohibitive.