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 to 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 $L^{0}$-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-priory 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, for synthetic and experimental data.

翻译：基于数据驱动的本构建模方法近年来受到广泛关注，其优势在于能天然融入物理与力学约束，规避传统唯象本构定律构建过程中耗时且复杂的难题，同时精准捕捉材料响应特征。然而，尽管基于神经网络的本构模型展现出优异的泛化能力，其大量可训练参数导致生成的表示形式难以实现可解释性。现有稀疏回归方法虽能获得可解释表达式，但需用户预先构建模型形式库，这种架构性设计本质上限制了模型表达性——其适用范围被局限于库中预设的函数形式。本研究提出采用平滑化$L^{0}$正则化训练物理增强型神经网络本构模型，旨在保留物理约束带来的可靠性，同时实现此前任何非预设模型形式的机器学习本构模型均未达成的可解释性目标。训练过程中，网络在拟合训练数据的同时，通过惩罚活跃参数数量实现网络稀疏化，并确保热力学一致性等本构约束条件。实验表明，该方法可针对可压缩/不可压缩超弹性材料、屈服函数及弹塑性硬化模型，基于合成数据与实验数据可靠获得兼具可解释性与可靠性的本构模型。