math.NAMar 27, 2026

Hyperelastic constitutive model discovery with differentiable finite elements and structure-preserving neural networks

Authors: Francesco Regazzoni

Organizations: MOX, Department of Mathematics, Politecnico di Milano, Milan, Italy

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Abstract

The discovery of constitutive laws from experimentally accessible measurements is a central problem in nonlinear computational mechanics. Many data-driven constitutive identification approaches rely either on paired strain-stress data or on full-field displacement measurements, both of which are difficult to obtain in realistic three-dimensional settings. We present a differentiable finite element framework for the discovery of hyperelastic material laws from partial observations, including boundary-only displacement measurements and global reaction forces. The method embeds the nonlinear finite element equilibrium problem directly into the learning loop, so that candidate strain-energy densities are assessed through the deformation fields and reactions they induce. This formulation enforces mechanical equilibrium as a constraint and allows the loss function to be evaluated only at observed locations. To ensure physical admissibility and promote numerical solvability throughout training, the constitutive response is represented by Hyperelastic Neural Networks, a structure-preserving neural class that enforces residual energy and stress-free conditions, frame indifference, isotropic material symmetry, polyconvexity, coercivity, and controlled volumetric growth by construction. The resulting PDE-constrained learning problem is solved using a quasi-Newton strategy combined with continuation and solver-aware backtracking. Numerical experiments in two- and three-dimensional finite elasticity demonstrate accurate recovery of hyperelastic isotropic responses from boundary-only data, robustness to measurement noise, and generalization across geometries, loading conditions, and boundary conditions.

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Figure 1: Overview of the Neural-DFEM framework. (a) HNNs constitute a family of neural networks designed to approximate the strain energy density functional of a hyperelastic isotropic material. By construction, an HNN satisfies seven physical and mathematical requirements that ensure existence of equilibria and physical consistency, as indicated in the figure alongside the architectural components that enforce them. The figure highlights the main design traits of HNNs, including weights with controlled sign, and a specific choice of the activation function. Optionally, HNNs can include skip connections, although they are not represented in the figure. (b) The training data consist of (possibly partial) displacement measurements and global reaction forces, coming from NexpN_{\mathrm{exp}} experiments. (c) Training is performed by embedding a differentiable finite element solver within the loss evaluation. The loss measures the discrepancy between predicted and observed data, while the Piola-Kirchhoff stress tensor is obtained by differentiating the HNN output with respect to the deformation gradient. (d) Once trained, the HNN can be integrated into standard finite element solvers to predict the deformation of new specimens, enabling generalization across geometries, loading conditions, and boundary conditions.
Figure 2: 2D setups. (a)–(b)–(c): Graphical representation of Setups 1–2–3 (left), and corresponding sample solutions obtained with the IH model (right). For Setup 3, six representative domains are shown. Light blue arrows represent displacement-controlled loads, whereas light orange arrows denote applied forces per unit surface. (d): Statistical distributions of the principal stretches (λ1,λ2)(\lambda_{1},\lambda_{2}) for the three setups considered. The scatter plots in the first column show the relative location, in the principal-stretch plane, of the deformation states obtained for the three setups, while the remaining three columns detail the principal-stretch distributions for each setup, displayed via kernel density estimation. For clarity of visualization, the principal stretches are plotted without enforcing an ordering, i.e. treating the two stretches symmetrically. Black lines correspond to canonical deformations described in Appendix D .
Figure 3: Results of 2D test cases (training on Setup 1, testing on Setup 2). (a) Boxplots of the displacement errors, normalized by the standard deviation of the ground-truth displacement, shown as a function of the applied load. The training data are generated using Setup 1 with σnoise=10−3\sigma_{\mathrm{noise}}=10^{-3} , while the test data correspond to Setup 2. Three approaches are compared: classical parameter fitting of a Neo-Hookean material; EUCLID [ 60 ] , based on the VFM; and the proposed Neural-DFEM. (b) Spatial distribution of the local displacement error for Setup 2 obtained with Neural-DFEM at the largest applied load (i.e., δ=1\delta=1 ) and with σnoise=10−3\sigma_{\mathrm{noise}}=10^{-3} , for the three material models considered. Note the different color scales used in the three cases.
Figure 4: Results of 2D test cases (training on Setup 1, testing on Setup 2 and Setup 3). (a) Predicted reaction force on the top boundary of Setup 2 as a function of the applied load and of the training data noise σnoise\sigma_{\mathrm{noise}} , compared with the reference reaction force obtained using the ground-truth material model. The insets show a magnified view of the force peak. (b) Boxplots of displacement and reaction force errors, both normalized by the standard deviation of the corresponding ground-truth, shown as a function of the training noise level σnoise∈{10−1,10−2,10−3,10−4,0}\sigma_{\mathrm{noise}}\in\{10^{-1},10^{-2},10^{-3},10^{-4},0\} . Results are reported for the three material models considered (indicated at the top of each column) and for both the full-field and boundary observation cases (see the legend in the bottom-right panel). The three rows correspond to Setups 1–2–3, respectively. Hence, while the first row shows the performance in the training setting (fitting regime), the second and third rows assess the ability of Neural-DFEM to generalize to mechanical setups different from those observed at training time. For Setup 3, no reaction force is defined, since no Dirichlet boundary conditions are prescribed.
Figure 5: 3D setups. (a)–(b)–(c): Graphical representation of Setups 4–5–6 (top), and corresponding sample solutions obtained with the IH model (bottom). Light blue arrows represent displacement-controlled loads, whereas light orange arrows denote applied forces per unit surface. (d): Statistical distributions of the principal stretches (λ1,λ2,λ3)(\lambda_{1},\lambda_{2},\lambda_{3}) for the three setups considered. The three scatter plots in the first row show the relative location, in principal-stretch space, of the deformation states obtained for the three setups, for the principal stretch triplet (λ1,λ2,λ3)(\lambda_{1},\lambda_{2},\lambda_{3}) , and for the (λ1,λ2)(\lambda_{1},\lambda_{2}) and (λ1,λ3)(\lambda_{1},\lambda_{3}) pairs, respectively. The second row shows the corresponding principal-stretch distributions for each setup, for the (λ1,λ2)(\lambda_{1},\lambda_{2}) and (λ1,λ3)(\lambda_{1},\lambda_{3}) pairs, displayed via kernel density estimation. For clarity of visualization, the principal stretches are plotted without enforcing an ordering, i.e. treating the stretches symmetrically. Black lines correspond to the canonical uniaxial tension or compression deformations.
Figure 6: Results of 3D test cases (training on Setup 4, testing on Setup 5 and Setup 6). (a) Predicted reaction force on the top boundary of Setup 5 as a function of the applied load and of the training data noise σnoise\sigma_{\mathrm{noise}} , compared with the reference reaction force obtained using the ground-truth material model. The insets show a magnified view of the force peak. (b) Boxplots of displacement and reaction force errors, both normalized by the standard deviation of the corresponding ground-truth, shown as a function of the training noise level σnoise∈{10−1,10−2,10−3,10−4,0}\sigma_{\mathrm{noise}}\in\{10^{-1},10^{-2},10^{-3},10^{-4},0\} , and of the observation type (full-fields vs boundary only, see legend). Results are reported for the three material models considered (indicated at the top of each column) and for both the full-field and boundary observation cases (see the legend in the bottom-right panel). The three rows correspond to Setups 4–5–6, respectively. Hence, while the first row shows the performance in the training setting (fitting regime), the second and third rows assess the ability of Neural-DFEM to generalize to mechanical setups different from those observed at training time. For Setup 6, no reaction force is defined, since no Dirichlet boundary conditions are prescribed.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

 2D specimens3D specimens
HyperparameterIHMRFUIHMRFU
Number of layers LL211112
Number of neurons per layer nin_{i}555555
Skip connections✗----✗
Isochoric input invariants✗✗✗✗✗✓
Initialization std. σinit\sigma_{\mathrm{init}}0.60.10.10.10.10.6
Appendix
Table 1: Selected hyperparameters for the HNN models.
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