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.

中文翻译：

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非线性模型中的参数确定性量化


从实验数据中估计模型参数是各个研究领域的常见做法。对于非线性模型，参数是使用最小化目标函数的优化算法来估计的。评估这些参数估计的确定性对于解决诸如“估计误差小于 5% 的概率是多少”、“我们的实验是否足够敏感以估计所有参数”以及“我们可以在准确拟合数据的同时改变每个参数多少”等问题至关重要。通常，使用模型的线性近似值来量化确定性水平。然而，我们表明，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获确定性水平。为了解决这些限制，我们提出了一种基于目标函数的 Hessian 近似的替代方法。我们表明，这种方法更准确地捕获了确定性水平，并且可以从几何上推导出来。我们通过涉及非线性超弹性材料本构模型的案例研究和在电解质溶液电导率非线性模型上的应用来证明我们方法的有效性。尽管计算成本较高，但当高度非线性模型中需要准确的确定性级别时，我们建议采用 Hessian 近似。