Constitutive model discovery refers to the task of identifying an appropriate model structure, usually from a predefined model library, while simultaneously inferring its material parameters. The data used for model discovery are measured in mechanical tests and are thus inevitably affected by noise which, in turn, induces uncertainties. Previously proposed methods for uncertainty quantification in model discovery either require the selection of a prior for the material parameters, are restricted to linear coefficients of the model library or are limited in the flexibility of the inferred parameter probability distribution. We therefore propose a partially Bayesian framework for uncertainty quantification in model discovery that does not require prior selection for the material parameters and also allows for the discovery of constitutive models with inner-non-linear parameters: First, we augment the available stress-deformation data with a Gaussian process. Second, we approximate the parameter distribution by a normalizing flow, which allows for modeling complex joint distributions. Third, we distill the parameter distribution by matching the distribution of stress-deformation functions induced by the parameters with the Gaussian process posterior. Fourth, we perform a Sobol' sensitivity analysis to obtain a sparse and interpretable model. We demonstrate the capability of our framework for both isotropic and experimental anisotropic data.

翻译：本构模型发现是指从预定义模型库中识别合适模型结构，并同时推断其材料参数的任务。用于模型发现的数据通过力学测试测得，因此不可避免地受到噪声影响，进而引入不确定性。先前提出的模型发现不确定性量化方法要么需要为材料参数选择先验分布，要么仅限于模型库的线性系数，要么在推断参数概率分布的灵活性方面存在局限。为此，我们提出了一种用于模型发现不确定性量化的部分贝叶斯框架，该框架无需为材料参数选择先验分布，同时能够发现具有内部非线性参数的本构模型：首先，我们使用高斯过程对可用的应力-变形数据进行增强。其次，我们通过归一化流近似参数分布，从而能够对复杂的联合分布进行建模。第三，我们通过使参数诱导的应力-变形函数分布与高斯过程后验相匹配来提炼参数分布。第四，我们进行Sobol'敏感性分析以获得稀疏且可解释的模型。我们通过各向同性和实验各向异性数据验证了该框架的有效性。