Non-smooth optimization meets automated material model discovery

2025-07-14

AI Summary

This work addresses the challenge of robust and efficient minimization of nonsmooth objective functions—specifically, (f(w) + alpha|w|_1)—in automated discovery of constitutive material models. We propose Path-ISTA, the first algorithm enabling exact and efficient computation of the (ell_1)-regularized solution path under nonquadratic loss functions. Unlike coordinate descent, LARS, or standard ISTA, Path-ISTA supports fully automatic hyperparameter selection and optimal sparse structure identification. We demonstrate its efficacy on uniaxial tension and pure shear experimental data, achieving end-to-end automated discovery of hyperelastic constitutive models. Results confirm superior convergence, robustness, and generalization in mechanical modeling. The method breaks from traditional, knowledge-intensive manual modeling paradigms, establishing a scalable and interpretable sparse optimization framework for data-driven materials modeling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• Robust minimization of non-smooth objective functions in material models
• Efficient computation of regularization paths for parameter selection
• Automated discovery of hyperelastic models from experimental data

Innovation

• Non-smooth L1-norm regularization for model discovery
• Coordinate descent and LARS for parameter selection
• Proximal gradient ISTA for non-quadratic functions