A Bayesian Approach for Data-Driven Dynamic Equation Discovery

North, Joshua S.; Wikle, Christopher K.; Schliep, Erin M.

doi:10.1007/s13253-022-00514-1

Cited by 11 publications

(18 citation statements)

References 76 publications

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“…While less common than the deterministic counterparts, methods to quantify uncertainty in the discovered equations have been proposed (Zhang & Lin, 2018;Niven et al, 2020;Yang et al, 2020;Fasel et al, 2022;North et al, 2022aNorth et al, , 2022bBhouri & Perdikaris, 2022). However, these methods generally do not account for uncertainty in the observed data, missing a vital piece of the statistical puzzle.…”

Section: Introductionmentioning

confidence: 99%

A Review of Data‐Driven Discovery for Dynamic Systems

North,

Wikle,

Schliep

2023

Int Statistical Rev

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SummaryMany real‐world scientific processes are governed by complex non‐linear dynamic systems that can be represented by differential equations. Recently, there has been an increased interest in learning, or discovering, the forms of the equations driving these complex non‐linear dynamic systems using data‐driven approaches. In this paper, we review the current literature on data‐driven discovery for dynamic systems. We provide a categorisation to the different approaches for data‐driven discovery and a unified mathematical framework to show the relationship between the approaches. Importantly, we discuss the role of statistics in the data‐driven discovery field, describe a possible approach by which the problem can be cast in a statistical framework and provide avenues for future work.

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Section: Introductionmentioning

confidence: 99%

A Review of Data‐Driven Discovery for Dynamic Systems

North,

Wikle,

Schliep

2023

Int Statistical Rev

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“…Bayesian approaches to spatial-temporal modeling have shown to be a versatile tool to capture complex phenomena and handle imbalanced or missing observations [1,2]. Among their wide range of applications are climate and weather modeling [3], disease mapping [4,5], medical image analysis [6,7], traffic management [8,9], environmental changes [10,11,12] and health economic evaluations [13].…”

Section: Introductionmentioning

confidence: 99%

Parallelized integrated nested Laplace approximations for fast Bayesian inference

Gaedke-Merzhäuser

Niekerk²,

Schenk³

et al. 2022

Stat Comput

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There is a growing demand for performing larger-scale Bayesian inference tasks, arising from greater data availability and higher-dimensional model parameter spaces. In this work we present parallelization strategies for the methodology of integrated nested Laplace approximations (INLA), a popular framework for performing approximate Bayesian inference on the class of Latent Gaussian models. Our approach makes use of nested thread-level parallelism, a parallel line search procedure using robust regression in INLA's optimization phase and the state-of-the-art sparse linear solver PARDISO. We leverage mutually independent function evaluations in the algorithm as well as advanced sparse linear algebra techniques. This way we can flexibly utilize the power of today's multi-core architectures. We demonstrate the performance of our new parallelization scheme on a number of different real-world applications. The introduction of parallelism leads to speedups of a factor 10 and more for all larger models. Our work is already integrated in the current version of the open-source R-INLA package, making its improved performance conveniently available to all users.

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“…This has been introduced in several papers, using combinations and polynoms of observed variables, as well as sparse regressions or model selection strategies (Brunton et al, 2016;Rudy et al, 2017;Mangiarotti and Huc, 2019). Those methods have then been extended to the case of noisy and irregular observation sampling, using Bayesian framework as in data assimilation (Bocquet et al, 2019;North et al, 2022). Alternatively, some authors used data assimilation and local linear regressions based on analogs (Tandeo et al, 2015;Lguensat et al, 2017), or iterative data assimilation coupled with neural networks (Brajard et al, 2020;Fablet et al, 2021), to make data-driven predictions without discovering equations.…”

Section: Introductionmentioning

confidence: 99%

Data-driven Reconstruction of Partially Observed Dynamical Systems

Ailliot

Sévellec

2022

Preprint

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Abstract. The state of the atmosphere, or of the ocean, cannot be exhaustively observed. Crucial parts might remain out of reach of proper monitoring. Also, defining the exact set of equations driving the atmosphere and ocean is virtually impossible because of their complexity. Hence, the goal of this paper is to obtain predictions of a partially observed dynamical system, without knowing the model equations. In this data-driven context, the article focuses on the Lorenz-63 system, where only the second and third components are observed, and access to the equations is not allowed. To account to those strong constraints, a combination of machine learning and data assimilation techniques is proposed. The key aspects are the following: the introduction of latent variables, a linear approximation of the dynamics, and a database that is updated iteratively, maximising the innovation likelihood. We find that the latent variables inferred by the procedure are related to the successive derivatives of the observed components of the dynamical system. The method is also able to reconstruct accurately the local dynamics of the partially observed system. Overall, the proposed methodology is simple, easy to code, and gives promising results, even in the case of small amounts of observations.

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A Bayesian Approach for Data-Driven Dynamic Equation Discovery

Cited by 11 publications

References 76 publications

A Review of Data‐Driven Discovery for Dynamic Systems

A Review of Data‐Driven Discovery for Dynamic Systems

Parallelized integrated nested Laplace approximations for fast Bayesian inference

Data-driven Reconstruction of Partially Observed Dynamical Systems

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