Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems

TL;DR

This paper tackles data assimilation for nonlinear dynamical systems by marrying INLA with iterative linearisation of nonlinear SPDE priors to produce a sequence of Gaussian Markov random field representations. At each iteration, a Gaussian approximation to the state is updated with INLA, and the linearisation point is refined via damped Gauss–Newton steps, yielding a Gauss–Newton–like optimisation for 4D-Var in the known-parameter case and a principled marginalisation over parameters when unknown. Empirically, iterated INLA achieves accurate state and parameter estimates and well-calibrated uncertainty across stochastic pendulum and PDE benchmarks, often outperforming EnKF/EnKS and other ML baselines while offering substantial computational savings over particle-based methods. The approach preserves interpretability by embedding physics in the prior and provides a flexible, uncertainty-aware alternative to existing DA methods, with limitations in scalability to very large-scale problems and potential extensions to non-Gaussian likelihoods.

Abstract

Data assimilation (DA) methods use priors arising from differential equations to robustly interpolate and extrapolate data. Popular techniques such as ensemble methods that handle high-dimensional, nonlinear PDE priors focus mostly on state estimation, however can have difficulty learning the parameters accurately. On the other hand, machine learning based approaches can naturally learn the state and parameters, but their applicability can be limited, or produce uncertainties that are hard to interpret. Inspired by the Integrated Nested Laplace Approximation (INLA) method in spatial statistics, we propose an alternative approach to DA based on iteratively linearising the dynamical model. This produces a Gaussian Markov random field at each iteration, enabling one to use INLA to infer the state and parameters. Our approach can be used for arbitrary nonlinear systems, while retaining interpretability, and is furthermore demonstrated to outperform existing methods on the DA task. By providing a more nuanced approach to handling nonlinear PDE priors, our methodology offers improved accuracy and robustness in predictions, especially where data sparsity is prevalent.

Figures (7)

• Figure 1: Comparison of the predictions made from a non-physics-informed model (Gaussian process regression / GPR) vs. a physics-informed model (iterated INLA). The ground truth is a simulation of the 1D Burgers' equation. Gray dots are observation locations.
• Figure 2: Comparison of the marginal state estimates $p(u_i | \boldsymbol{y})$ on the pendulum experiment. We display the credible intervals (CI) in blue shades; black dots are noisy observations from a sample simulation, displayed in orange. For methods (b)--(d), we display the maximum mean discrepancy (MMD) from the SMC result (a), which we take as the gold standard. Iterated INLA performs best both qualitatively and in terms of the MMD score.
• Figure 3: Estimated marginal posterior densities for the $b$ parameter and the system noise parameter $\sigma_u$ using (a) iINLA-II and (b) EnKS for the pendulum experiment. The marginal distributions are displayed in orange on the respective axes. We also plot the marginal distributions obtained by SMC in blue. For the $\sigma_u$ parameter, the estimates obtained by EnKS diverges from SMC, while iterated INLA recovers it correctly.
• Figure 4: Comparison of the predicted standard deviations on the Allen-Cahn example. The predictions are generally underconfident for iINLA due to the presence of $\sigma_u > 0$. Gray dots are observation locations.
• Figure 5: Results on the Burgers' experiment

Theorems & Definitions (9)

• Remark 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• proof
• proof
• Definition D.1: lototsky2017stochastic, Definition 3.2.10