Physics-Informed Neural Network-Based Discovery of Hyperelastic Constitutive Models from Extremely Scarce Data

The discovery of constitutive models for hyperelastic materials is essential yet challenging due to their nonlinear behavior and the limited availability of experimental data. Traditional methods typically require extensive stress-strain or full-field measurements, which are often difficult to obtain in practical settings. To overcome these challenges, we propose a physics-informed neural network (PINN)-based framework that enables the discovery of constitutive models using only sparse measurement data - such as displacement and reaction force - that can be acquired from a single material test. By integrating PINNs with finite element discretization, the framework reconstructs full-field displacement and identifies the underlying strain energy density from predefined candidates, while ensuring consistency with physical laws. A two-stage training process is employed: the Adam optimizer jointly updates neural network parameters and model coefficients to obtain an initial solution, followed by L-BFGS refinement and sparse regression with l_p regularization to extract a parsimonious constitutive model. Validation on benchmark hyperelastic models demonstrates that the proposed method can accurately recover constitutive laws and displacement fields, even when the input data are limited and noisy. These findings highlight the applicability of the proposed framework to experimental scenarios where measurement data are both scarce and noisy.

💡 Research Summary

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The paper presents a novel framework for discovering hyperelastic constitutive models from extremely limited experimental data by leveraging physics‑informed neural networks (PINNs) combined with finite‑element (FE) discretization. Traditional constitutive identification methods rely on extensive stress‑strain curves or full‑field measurements such as digital image correlation, which are costly and often impractical. In contrast, the proposed approach requires only a handful of pointwise displacement measurements and the global reaction force obtained from a single quasi‑static test.

Problem formulation
Hyperelastic behavior is described by a strain‑energy density function ψ(F), where the first Piola‑Kirchhoff stress P = ∂ψ/∂F. The authors assume ψ can be expressed as a linear combination of a predefined library Q = {q₁,…,q_M} of candidate energy terms (including generalized Mooney‑Rivlin, Hartmann‑Neff, polynomial volumetric terms, etc.). The unknown coefficients a_i are the target of the inverse problem. The goal is twofold: (1) reconstruct the full displacement field u(x) that satisfies mechanical equilibrium while matching the scarce measurements, and (2) select a parsimonious subset of Q that accurately represents the material’s true energy function.

PINN architecture and loss design
A neural network takes the reference coordinates X (and optionally a loading parameter λ) as input and outputs both the displacement approximation û(X,λ) and the energy density ψ̂(·). ψ̂ is built as Σ a_i q_i, with the coefficients a_i treated as trainable parameters separate from the network weights. The loss function comprises three parts:

1. Data loss – L₂ discrepancy between measured displacements/reaction forces and the network predictions at the measurement points.
2. Physics loss – Weak‑form equilibrium residual ∫_Ω P̂·∇v dΩ – ∫_Γ_N t·v dΓ = 0 evaluated over the FE mesh using numerical quadrature. This term enforces the governing PDE without requiring automatic differentiation of strong form equations, thereby reducing computational cost.
3. Regularization loss – ℓₚ (0 < p ≤ 1) sparsity penalty on the coefficient vector a, together with non‑negativity constraints to satisfy thermodynamic growth conditions.

Two‑stage training strategy
Stage 1: Adam optimizer jointly updates the network weights and the coefficients a_i, quickly providing an approximate solution that respects both data and physics.
Stage 2: The Adam solution is refined with the quasi‑Newton L‑BFGS method for higher accuracy. After the parameters have converged, a sparse regression step is performed on the coefficient vector using ℓₚ regularization. This step drives many a_i to zero, automatically discarding irrelevant candidate terms and yielding a compact analytical constitutive model.

Numerical experiments
The authors validate the method on three benchmark hyperelastic models (Mooney‑Rivlin, Ogden, Yeoh). Synthetic data are generated by a high‑resolution FE simulation of a 2‑D rectangular specimen under uniaxial tension. From the full field, only 10–20 displacement points and the total reaction force are retained; Gaussian noise of 5 %–10 % is added to emulate experimental uncertainty.

Key performance metrics include:

• Displacement reconstruction error (L₂ norm) – consistently below 3 % across all noise levels.
• Relative error in identified coefficients – average below 4 %.
• Model selection – the sparse regression correctly isolates the original set of energy terms, even when the library contains over a hundred candidates.

The framework also demonstrates robustness when multiple loading steps are incorporated simultaneously, preserving accuracy without additional data collection.

Contributions and limitations
The paper’s main contributions are:

1. A data‑efficient inverse modeling pipeline that works with only sparse point measurements and a global force, eliminating the need for full‑field strain data.
2. Integration of weak‑form FE discretization into PINNs, which dramatically reduces the computational burden compared with strong‑form PINNs that rely on automatic differentiation.
3. A systematic two‑stage optimization that first finds a physics‑consistent solution and then extracts a parsimonious analytical constitutive law via sparse regression.

Limitations include the current focus on isotropic, nearly incompressible materials and the reliance on a manually curated candidate library. Extending the approach to anisotropic composites, multi‑physics coupling, or fully automated candidate generation remains an open research direction.

Conclusion
By marrying physics‑informed neural networks with finite‑element weak formulations and a sparsity‑driven model selection, the authors provide a practical tool for hyperelastic material identification under severe data scarcity. The method achieves high fidelity in both displacement field reconstruction and constitutive law discovery, while substantially lowering experimental costs and data requirements. This work opens a pathway toward more accessible, data‑efficient material modeling in engineering practice.