Non-smooth optimization meets automated material model discovery

Automated material model discovery disrupts the tedious and time-consuming cycle of iteratively calibrating and modifying manually designed models. Non-smooth L1-norm regularization is the backbone of automated model discovery; however, the current literature on automated material model discovery offers limited insights into the robust and efficient minimization of non-smooth objective functions. In this work, we examine the minimization of functions of the form f(w) + a ||w||_1, where w are the material model parameters, f is a metric that quantifies the mismatch between the material model and the observed data, and a is a regularization parameter that determines the sparsity of the solution. We investigate both the straightforward case where f is quadratic and the more complex scenario where it is non-quadratic or even non-convex. Importantly, we do not only focus on methods that solve the sparse regression problem for a given value of the regularization parameter a, but propose methods to efficiently compute the entire regularization path, facilitating the selection of a suitable a. Specifically, we present four algorithms and discuss their roles for automated material model discovery in mechanics: First, we recapitulate a well-known coordinate descent algorithm that solves the minimization problem assuming that f is quadratic for a given value of a, also known as the LASSO. Second, we discuss the algorithm LARS, which automatically determines the critical values of a, at which material parameters in w are set to zero. Third, we propose to use the proximal gradient method ISTA for automated material model discovery if f is not quadratic, and fourth, we suggest a pathwise extension of ISTA for computing the regularization path. We demonstrate the applicability of all algorithms for the discovery of hyperelastic material models from uniaxial tension and simple shear data.