The internal law of a material can be discovered from its boundary

Abstract

Since the earliest stages of human civilization, advances in technology have been tightly linked to our ability to understand and predict the mechanical behavior of materials. In recent years, this challenge has increasingly been framed within the broader paradigm of data-driven scientific discovery, where governing laws are inferred directly from observations. However, existing methods require either stress-strain pairs or full-field displacement measurements, which are often inaccessible in practice. We introduce Neural-DFEM, a method that enables unsupervised discovery of hyperelastic material laws even from partial observations, such as boundary-only measurements. The method embeds a differentiable finite element solver within the learning loop, directly linking candidate energy functionals to available measurements. To guarantee thermodynamic consistency and mathematical well-posedness throughout training, the method employs Hyperelastic Neural Networks, a novel structure-preserving neural architecture that enforces frame indifference, material symmetry, polyconvexity, and coercivity by design. The resulting framework enables robust material model discovery in both two- and three-dimensional settings, including scenarios with boundary-only measurements. Neural-DFEM allows for generalization across geometries and loading conditions, and exhibits unprecedented accuracy and strong resilience to measurement noise. Our results demonstrate that reliable identification of material laws is achievable even under partial observability when strong physical inductive biases are embedded in the learning architecture.

Figures (6)

• Figure 1: Overview of the Neural-DFEM framework. (a) HNNs constitute a novel family of neural networks designed to approximate the strain energy density functional of a hyperelastic isotropic material. By construction, an HNN satisfies the seven physical and mathematical requirements necessary for well-posedness 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 $N_{\mathrm{exp}}$ 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 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 \ref{['sec:canonical_deformations']}.
• Figure 3: 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 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 $(\lambda_1,\lambda_2,\lambda_3)$, and for the $(\lambda_1,\lambda_2)$ and $(\lambda_1,\lambda_3)$ pairs, respectively. The second row shows the corresponding principal-stretch distributions for each setup, for the $(\lambda_1,\lambda_2)$ and $(\lambda_1,\lambda_3)$ 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 4: 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 $\sigma_{\mathrm{noise}} = 10^{-3}$, while the test data correspond to Setup 2. Three approaches are compared: classical parameter fitting of a Neo-Hookean material; EUCLID thakolkaran2022nn, 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., $\delta = 1$) and with $\sigma_{\mathrm{noise}} = 10^{-3}$, for the three material models considered. Note the different color scales used in the three cases.
• Figure 5: 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 $\sigma_{\mathrm{noise}}$, 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 $\sigma_{\mathrm{noise}} \in \{ 10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}, 0\}$. 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.

Theorems & Definitions (15)

• Theorem 1
• proof
• Remark 1
• Definition 1
• Theorem 2
• proof
• Definition 2
• Theorem 3
• proof
• Remark 2