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.

利用神经网络的数据驱动本构模型，在加入物理和机制约束的同时，通过训练及 $l^0$ 正则化的方法，实现了可解释性和可信赖性的简约化表征，适用于可压缩与不可压缩高弹性、屈服函数和强化模型的合成和实验数据。

物理增强神经网络的极度稀疏化方法用于机械学中可解释的模型发现

Extreme sparsification of physics-augmented neural networks for  interpretable model discovery in mechanics

Data-driven constitutive modeling frameworks based on neural networks and
classical representation theorems have recently gained considerable attention
due to their ability to easily incorporate constitutive constraints and their
excellent generalization performance. In these models, the stress prediction
follows from a linear combination of invariant-dependent coefficient functions
and known tensor basis generators. However, thus far the formulations have been
limited to stress representations based on the classical Rivlin and Ericksen
form, while the performance of alternative representations has yet to be
investigated. In this work, we survey a variety of tensor basis neural network
models for modeling hyperelastic materials in a finite deformation context,
including a number of so far unexplored formulations which use theoretically
equivalent invariants and generators to Finger-Rivlin-Ericksen. Furthermore, we
compare potential-based and coefficient-based approaches, as well as different
calibration techniques. Nine variants are tested against both noisy and
noiseless datasets for three different materials. Theoretical and practical
insights into the performance of each formulation are given.

基于数据驱动的神经网络和经典表示定理的本构建模框架因其易于整合本构约束和卓越的泛化性能而受到广泛关注。本研究通过测试多种不同张量基生成器的模型，对有限变形情况下的高弹性材料进行建模，包括一些尚未探索使用与 Finger-Rivlin-Ericksen 理论等效的不变量和生成器的公式。此外，我们比较了基于势函数和系数的方法以及不同的校准技术，并对三种不同材料的噪声和无噪声数据集进行了九个变体的测试，提供了每种公式性能的理论和实用洞察。