Iterated INLA for State and Parameter Estimation in Nonlinear Dynamical Systems

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.

• The paper introduces an iterative INLA method that integrates state and parameter estimation via successive linearization of nonlinear models.
• It transforms nonlinear PDE priors into Gaussian Markov Random Fields, enabling efficient application of INLA for improved inference.
• Practical experiments demonstrate enhanced prediction accuracy and robustness compared to traditional data assimilation techniques.

Iterated INLA for Enhanced Data Assimilation in Nonlinear Dynamical Systems

Introduction

In traditional approaches to data assimilation (DA) in systems governed by nonlinear partial differential equations (PDE), challenges often arise due to the computational complexity and the difficulty in accurately estimating model parameters. Ensemble methods and machine learning-based alternatives each offer partial solutions, yet they either struggle with parameter estimation or entail interpretability issues concerning uncertainties. This paper introduces an advanced DA methodology leveraging Integrated Nested Laplace Approximation (INLA), specifically designed for arbitrary nonlinear systems. By iteratively linearizing the dynamical model to form a Gaussian Markov Random Field (GMRF), the method succinctly integrates state and parameter inference, thereby providing a significant improvement in prediction accuracy and robustness, particularly in data-sparse scenarios.

Methodology

The core of our approach revolves around the SPDE-INLA technique, extending it to suit nonlinear PDE priors via iterative linearization. This results in a direct representation of the GP priors as GMRFs, onto which INLA is applied for efficient state and parameter estimation. The method asserts two primary strengths: the ability to employ priors derived from any nonlinear PDE and to furnish accurate estimates of non-Gaussian state and posterior distributions. By also addressing model uncertainty and incorporating expert knowledge, our approach significantly enhances interpretability and predictability in DA tasks.

Theoretical Implications

Our methodology's theoretical foundation capitalizes on the elegant framework provided by INLA for latent Gaussian models. Through the iterative linearization of the nonlinear dynamical models, we construct an adaptable framework that resonates well with the inherent complexities of nonlinear PDE systems. The feasibility of extending this approach to handle non-Gaussian likelihoods and exploring nested Laplace approximations presents a promising avenue for future research. Additionally, the convergence of our algorithm to the solutions provided by the Gauß-Newton method reinforces our method's reliability and effectiveness.

Practical Implications

Practically, the implementation of our iterative INLA method showcases its utility in accurately recovering states and parameters from sparse and noisy data across various nonlinear PDE systems. The experiments conducted illustrate the method's robustness and its superior performance against established DA techniques. Especially in scenarios where traditional methods face numerical stability issues or potential overfitting due to incorrect model assumptions, our method exhibits remarkable resilience and consistency. Its application extends potential benefits to fields reliant on complex dynamical models, including environmental sciences, engineering, and applied physics.

Future Prospects

Looking ahead, the development of more scalable versions of the proposed methodology could address current limitations regarding high-dimensional problems. Integrating the method with filtering/smoothing strategies specific to temporal data may also expedite computations and enhance performance. The exploration into non-Gaussian likelihoods and more complex model dynamics, including stochastic components and multi-scale phenomena, will likely broaden the method's applicability. As the method matures, its contributions to improving model predictions, understanding system uncertainties, and guiding empirical studies could be substantial.

Conclusion

In summation, the proposed iterated INLA method provides an innovative and theoretically sound approach to the complex challenges of DA in nonlinear dynamical systems. Its ability to combine accurate state and parameter estimation with improved computational efficiency and robustness marks a significant advancement in the field. As we continue to explore its extensions and applications, the full potential of this method in contributing to various scientific and engineering disciplines remains promising and widely anticipated.