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.

翻译：数据同化方法利用微分方程产生的先验信息，能够稳健地插值和外推数据。诸如处理高维非线性偏微分方程先验的集合方法等主流技术，主要聚焦于状态估计，但在精确学习参数方面存在困难。另一方面，基于机器学习的方法虽能自然学习状态和参数，但其适用性可能受限，或产生难以解释的不确定性。受空间统计学中集成嵌套拉普拉斯近似方法的启发，我们提出了一种基于动力学模型迭代线性化的数据同化替代方案。该方法在每次迭代中生成高斯马尔可夫随机场，从而能够利用INLA推断状态和参数。我们的方法适用于任意非线性系统，同时保持可解释性，并在数据同化任务中证明优于现有方法。通过提供更精细的非线性偏微分方程先验处理策略，本方法在预测精度和鲁棒性方面均有提升，尤其适用于数据稀疏场景。