  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.
