# Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields

arXiv: https://arxiv.org/abs/2310.03745 · HTML: https://mlsanity.com/paper/2310.03745/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 77 references that are in the map.
- First posted on arXiv 11 Sep 2023.

## Builds on (most cited first, 30 of 33)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Score-Based Generative Models Detect Manifolds](https://mlsanity.com/paper/2206.01018/) (2022)
- [Adversarial score matching and improved sampling for image generation](https://mlsanity.com/paper/2009.05475/) (2020)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Bayesian Hypernetworks](https://mlsanity.com/paper/1710.04759/) (2017)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [Neural Diffusion Processes](https://mlsanity.com/paper/2206.03992/) (2022)
- [Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity](https://mlsanity.com/paper/2001.04292/) (2020)
- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [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)
- [Automated discovery of generalized standard material models with EUCLID](https://mlsanity.com/paper/2211.04453/) (2022)
- [NN-EUCLID: deep-learning hyperelasticity without stress data](https://mlsanity.com/paper/2205.06664/) (2022)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [Diffusion Probabilistic Fields](https://mlsanity.com/paper/2303.00165/) (2023)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Review](https://mlsanity.com/paper/2303.07647/) (2023)
- [Data-driven Modeling of the Mechanical Behavior of Anisotropic Soft Biological Tissue](https://mlsanity.com/paper/2107.05388/) (2021)
- [A comparative study on different neural network architectures to model inelasticity](https://mlsanity.com/paper/2303.03402/) (2023)
- [Manifold Diffusion Fields](https://mlsanity.com/paper/2305.15586/) (2023)
- [Predicting the Mechanical Properties of Biopolymer Gels Using Neural Networks Trained on Discrete Fiber Network Data](https://mlsanity.com/paper/2101.11712/) (2021)
- [Fast characterization of inducible regions of atrial fibrillation models with multi-fidelity Gaussian process classification](https://mlsanity.com/paper/2112.08075/) (2021)

## Built on by (most cited first, 7 of 7)

- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches](https://mlsanity.com/paper/2503.00575/) (2025)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Unsupervised full-field Bayesian inference of orthotropic hyperelasticity from a single biaxial test: a myocardial case study](https://mlsanity.com/paper/2510.09498/) (2025)
- [Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields](https://mlsanity.com/paper/2506.08146/) (2025)
- [Generative AI for material design: A mechanics perspective from burgers to matter](https://mlsanity.com/paper/2604.03409/) (2026)
- [CANN-EUCLID: unsupervised constitutive artificial neural network model discovery from full-field data](https://mlsanity.com/paper/2606.14565/) (2026)

## Abstract (arXiv)

Many natural materials exhibit highly complex, nonlinear, anisotropic, and heterogeneous mechanical properties. Recently, it has been demonstrated that data-driven strain energy functions possess the flexibility to capture the behavior of these complex materials with high accuracy while satisfying physics-based constraints. However, most of these approaches disregard the uncertainty in the estimates and the spatial heterogeneity of these materials. In this work, we leverage recent advances in generative models to address these issues. We use as building block neural ordinary equations (NODE) that -- by construction -- create polyconvex strain energy functions, a key property of realistic hyperelastic material models. We combine this approach with probabilistic diffusion models to generate new samples of strain energy functions. This technique allows us to sample a vector of Gaussian white noise and translate it to NODE parameters thereby representing plausible strain energy functions. We extend our approach to spatially correlated diffusion resulting in heterogeneous material properties for arbitrary geometries. We extensively test our method with synthetic and experimental data on biological tissues and run finite element simulations with various degrees of spatial heterogeneity. We believe this approach is a major step forward including uncertainty in predictive, data-driven models of hyperelasticity