  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.
