Physics-Informed Reduced-Order Operator Learning for Hyperelasticity in Continuum Micromechanics

TL;DR

This paper introduces a physics-informed reduced-order operator learning framework for hyperelasticity in continuum micromechanics, significantly reducing computational costs while maintaining accuracy in stress predictions.

Contribution

It combines EquiNO with Q-DEIM to enable efficient training and inference of stress fields in three-dimensional hyperelastic RVEs, improving computational speed and scalability.

Findings

Q-DEIM reduces training cost by about three orders of magnitude.

The method achieves speed-up factors of 10^3 to 10^4 over full-field computations.

Prediction accuracy improves with more offline snapshots, effectively interpolating and extrapolating stress fields.

Abstract

Physics-informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite-element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neural Operator (EquiNO) with the QR-based discrete empirical interpolation method (Q-DEIM). EquiNO learns only the modal coefficients of reduced displacement-fluctuation and first Piola-Kirchhoff stress representations built from periodic and divergence-free bases, thereby enforcing periodicity and mechanical equilibrium by construction. Q-DEIM then identifies a small set of spatial points through a column-pivoted QR factorization of the stress basis and restricts constitutive evaluations during training to these points alone. This makes full-batch second-order optimization practical for three-dimensional representative volume elements (RVEs). Homogenized first Piola-Kirchhoff stresses are recovered directly from the offline-averaged reduced stress modes, without the need to reconstruct the full stress field at inference time. We validate the framework on two three-dimensional finite-strain hyperelastic RVEs. Q-DEIM reduces the per-step training cost by roughly three orders of magnitude relative to full-field loss evaluation, while reduced homogenization achieves speed-up factors of order $10^3$ to $10^4$ over direct full-field computations. Despite relying on only a small number of offline snapshot loading paths for basis construction, the method accurately interpolates and extrapolates both microscopic stress fields and homogenized stresses, with prediction quality improving systematically as more snapshots are added.