We propose a new approach for data-driven automated discovery of hyperelastic constitutive laws. The approach is unsupervised, i.e., it requires no stress data but only displacement and global force data, which are realistically available through mechanical testing and digital image correlation techniques; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a large catalogue of candidate functions; it is one-shot, i.e., discovery only needs one experiment - but can use more if available. The problem of unsupervised discovery is solved by enforcing equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity of the solution is achieved by l_p regularization combined with thresholding, which calls for a non-linear optimization scheme. The ensuing fully automated algorithm leverages physics-based constraints for the automatic determination of the penalty parameter in the regularization term. Using numerically generated data including artificial noise, we demonstrate the ability of the approach to accurately discover five hyperelastic models of different complexity. We also show that, if a ``true'' feature is missing in the function library, the proposed approach is able to surrogate it in such a way that the actual response is still accurately predicted.

中文翻译：

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无监督发现可解释的超弹性本构定律

我们提出了一种数据驱动的自动发现超弹性本构定律的新方法。该方法是无监督的，即它不需要应力数据，只需要位移和全局力数据，这些数据可以通过机械测试和数字图像相关技术实际获得；它提供了可解释的模型，即由通过大量候选函数的稀疏回归发现的简约数学表达式体现的模型；它是一次性的，即发现只需要一个实验——但如果有的话可以使用更多。无监督发现的问题是通过在域的整体和加载边界处强制执行平衡约束来解决的。解决方案的稀疏性是通过 l_p 正则化结合阈值来实现的，这需要非线性优化方案。随后的全自动算法利用基于物理的约束来自动确定正则化项中的惩罚参数。使用包括人工噪声在内的数值生成数据，我们展示了该方法准确发现五种不同复杂性的超弹性模型的能力。我们还表明，如果函数库中缺少“真实”特征，则所提出的方法能够以这样一种方式替代它，从而仍然可以准确预测实际响应。我们展示了该方法准确发现五个不同复杂度的超弹性模型的能力。我们还表明，如果函数库中缺少“真实”特征，则所提出的方法能够以这样一种方式替代它，从而仍然可以准确预测实际响应。我们展示了该方法准确发现五个不同复杂度的超弹性模型的能力。我们还表明，如果函数库中缺少“真实”特征，则所提出的方法能够以这样一种方式替代它，从而仍然可以准确预测实际响应。