Unsupervised Constitutive Model Discovery from Sparse and Noisy Data

TL;DR

This work addresses unsupervised constitutive model discovery from sparse, noisy displacement data by coupling the statistical finite element method (statFEM) with the EUCLID framework. It combines Bayesian state assimilation of measurements (via a nonintrusive polynomial chaos forward model) with sparse-regression model discovery in a weak form to enforce equilibrium and identify parsimonious constitutive laws. The approach demonstrates improved robustness and accuracy for isotropic hyperelastic materials, recovering Neo-Hookean and Mooney–Rivlin energy functions even with few sensors and significant noise. The framework offers a practical route to data-driven, physically interpretable material models under realistic sensing constraints, with potential applicability beyond the tested isotropic cases.

Abstract

Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations or aim to yield interpretable constitutive models. In this work, we couple statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM--EUCLID. The framework is demonstrated for isotropic hyperelastic materials. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws.

Figures (12)

• Figure 1: Graphical illustration of the iterative model discovery framework. The loop iteratively updates the initial model predictions based on observational data via statFEM and discovers a constitutive model until convergence is reached. The dashed line shows the displacement assimilation process without updating the constitutive model. This framework is adapted from Arendt2012.
• Figure 2: Pareto analysis of the model discovery problem \ref{['eq:PaperC_discovery_exntedned']}. (a) Full path. (b) Zoomed view for $\text{RMSE}_{\lambda} < \tau$.
• Figure 3: Geometry and loading of the plate with a central hole.
• Figure 4: Meshing and sensor locations for different sensor counts. The black element on the bottom left is the representative element, for which we show the strain energy density $W$ in \ref{['fig:plateWithHole_energy_vs_f']}, \ref{['fig:plateWithHole_energy_vs_J_e3']}, and \ref{['fig:plateWithHole_energy_vs_J_e4']}.
• Figure 5: Convergence of the relative strain energy reconstruction error $\epsilon_W$ over model discovery iterations for $\sigma_{ \mathbf{e} }=10^{-4}$. (a) Full iteration history. (b) Zoomed view for iterations $k \geq 3$ to highlight convergence stability.