Data-driven constitutive modeling with neural networks has received increased interest in recent years due to its ability to easily incorporate physical and mechanistic constraints and to overcome the challenging and time-consuming task of formulating phenomenological constitutive laws that can accurately capture the observed material response. However, even though neural network-based constitutive laws have been shown to generalize proficiently, the generated representations are not easily interpretable due to their high number of trainable parameters. Sparse regression approaches exist that allow for obtaining interpretable expressions, but the user is tasked with creating a library of model forms which by construction limits their expressiveness to the functional forms provided in the libraries. In this work, we propose to train regularized physics-augmented neural network-based constitutive models utilizing a smoothed version of -regularization. This aims to maintain the trustworthiness inherited by the physical constraints, but also enables interpretability which has not been possible thus far on any type of machine learning-based constitutive model where model forms were not assumed a priori but were actually discovered. During the training process, the network simultaneously fits the training data and penalizes the number of active parameters, while also ensuring constitutive constraints such as thermodynamic consistency. We show that the method can reliably obtain interpretable and trustworthy constitutive models for compressible and incompressible hyperelasticity, yield functions, and hardening models for elastoplasticity, using synthetic and experimental data. This work aims to set a new paradigm for interpretable machine learning models in the broad area of solid mechanics where low and limited data is available along with prior knowledge of physical constraints that the learned maps need to obey. This paradigm can potentially be extended to a broader spectrum of scientific exploration.

中文翻译：

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用于力学中可解释模型发现的物理增强神经网络的极端稀疏化

近年来，利用神经网络进行数据驱动的本构建模受到了越来越多的关注，因为它能够轻松地结合物理和机械约束，并克服制定现象学本构定律的挑战性和耗时的任务，从而准确地捕获观察到的材料响应。然而，尽管基于神经网络的本构定律已被证明可以有效地泛化，但由于其大量可训练参数，生成的表示并不容易解释。存在允许获得可解释表达式的稀疏回归方法，但用户的任务是创建模型形式库，该库通过构造将其表达能力限制为库中提供的函数形式。在这项工作中，我们建议利用平滑版本的正则化来训练基于正则化物理增强神经网络的本构模型。这样做的目的是保持物理约束所继承的可信度，同时也实现了可解释性，这是迄今为止在任何类型的基于机器学习的本构模型上不可能实现的，其中模型形式不是先验假定的，而是实际发现的。在训练过程中，网络同时拟合训练数据并惩罚活跃参数的数量，同时还保证热力学一致性等本构约束。我们证明，利用合成和实验数据，该方法可以可靠地获得可压缩和不可压缩超弹性、屈服函数和弹塑性硬化模型的可解释且值得信赖的本构模型。这项工作的目的是在固体力学的广泛领域为可解释的机器学习模型建立一个新的范例，其中可以使用少量且有限的数据以及所学习的图需要遵守的物理约束的先验知识。这种范式有可能扩展到更广泛的科学探索领域。