Parameter certainty quantification in nonlinear models

IF 5.7 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY International Journal of Engineering Science Pub Date : 2024-11-04 DOI:10.1016/j.ijengsci.2024.104163
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Abstract

Estimating model parameters from experimental data is a common practice across various research fields. For nonlinear models, the parameters are estimated using an optimization algorithm that minimizes an objective function. Assessing the certainty of these parameter estimates is crucial to address questions such as “what is the probability the estimation error is smaller than 5%?”, “is our experiment sensitive enough to estimate all parameters?”, and “how much can we change each parameter while still fitting the data accurately?”. Typically, the certainty levels are quantified using a linear approximation of the model. However, we show that in models that are highly nonlinear in their parameters or in the presence of large experimental errors, this method fails to capture the certainty levels accurately. To address these limitations, we present an alternative method based on the Hessian approximation of the objective function. We show that this method captures the certainty levels more accurately and can be derived geometrically. We demonstrate the efficacy of our approach through a case study involving a nonlinear hyperelastic material constitutive model and an application on a nonlinear model for the conductivity of electrolyte solutions. Despite its higher computational cost, we recommend adopting the Hessian approximation when accurate certainty levels are required in highly nonlinear models.
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非线性模型中的参数确定性量化
根据实验数据估算模型参数是各个研究领域的常见做法。对于非线性模型,参数估计采用优化算法,使目标函数最小化。评估这些参数估计的确定性对于解决以下问题至关重要,例如 "估计误差小于 5%的概率是多少?"、"我们的实验是否足够敏感,可以估计出所有参数?"以及 "我们可以在精确拟合数据的同时改变每个参数多少?"。通常情况下,确定性水平是通过模型的线性近似来量化的。然而,我们的研究表明,在参数高度非线性或存在较大实验误差的模型中,这种方法无法准确捕捉确定性水平。为了解决这些局限性,我们提出了一种基于目标函数 Hessian 近似值的替代方法。我们证明,这种方法能更准确地捕捉确定性水平,并能以几何方式推导出来。我们通过涉及非线性超弹性材料构成模型的案例研究和电解质溶液电导率非线性模型的应用,证明了我们方法的有效性。尽管计算成本较高,但当高度非线性模型需要精确的确定性水平时,我们建议采用赫塞斯近似法。
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来源期刊
International Journal of Engineering Science
International Journal of Engineering Science 工程技术-工程:综合
CiteScore
11.80
自引率
16.70%
发文量
86
审稿时长
45 days
期刊介绍: The International Journal of Engineering Science is not limited to a specific aspect of science and engineering but is instead devoted to a wide range of subfields in the engineering sciences. While it encourages a broad spectrum of contribution in the engineering sciences, its core interest lies in issues concerning material modeling and response. Articles of interdisciplinary nature are particularly welcome. The primary goal of the new editors is to maintain high quality of publications. There will be a commitment to expediting the time taken for the publication of the papers. The articles that are sent for reviews will have names of the authors deleted with a view towards enhancing the objectivity and fairness of the review process. Articles that are devoted to the purely mathematical aspects without a discussion of the physical implications of the results or the consideration of specific examples are discouraged. Articles concerning material science should not be limited merely to a description and recording of observations but should contain theoretical or quantitative discussion of the results.
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