# Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields

arXiv: https://arxiv.org/abs/2506.08146 · HTML: https://mlsanity.com/paper/2506.08146/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 59 references that are in the map.
- First posted on arXiv 9 Jun 2025.

## Builds on (most cited first, 11 of 11)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural networks meet hyperelasticity: A guide to enforcing physics](https://mlsanity.com/paper/2302.02403/) (2023)
- [JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science](https://mlsanity.com/paper/2212.00964/) (2022)
- [Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors](https://mlsanity.com/paper/2107.02926/) (2021)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling](https://mlsanity.com/paper/2109.11028/) (2021)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networks](https://mlsanity.com/paper/2402.10741/) (2024)
- [Improving the performance of Stein variational inference through extreme sparsification of physically-constrained neural network models](https://mlsanity.com/paper/2407.00761/) (2024)
- [Inference of Heterogeneous Material Properties via Infinite-Dimensional Integrated DIC](https://mlsanity.com/paper/2408.10217/) (2024)

## Abstract (arXiv)

We propose a new framework for identifying mechanical properties of heterogeneous materials without a closed-form constitutive equation. Given a full-field measurement of the displacement field, for instance as obtained from digital image correlation (DIC), a continuous approximation of the strain field is obtained by training a neural network that incorporates Fourier features to effectively capture sharp gradients in the data. A physics-based data-driven method built upon ordinary neural differential equations (NODEs) is employed to discover constitutive equations. The NODE framework can represent arbitrary materials while satisfying constraints in the theory of constitutive equations by default. To account for heterogeneity, a hyper-network is defined, where the input is the material coordinate system, and the output is the NODE-based constitutive equation. The parameters of the hyper-network are optimized by minimizing a multi-objective loss function that includes penalty terms for violations of the strong form of the equilibrium equations of elasticity and the associated Neumann boundary conditions. We showcase the framework with several numerical examples, including heterogeneity arising from variations in material parameters, spatial transitions from isotropy to anisotropy, material identification in the presence of noise, and, ultimately, application to experimental data. As the numerical results suggest, the proposed approach is robust and general in identifying the mechanical properties of heterogeneous materials with very few assumptions, making it a suitable alternative to classical inverse methods.