# On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

arXiv: https://arxiv.org/abs/2109.11028 · HTML: https://mlsanity.com/paper/2109.11028/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2021.

## Built on by (most cited first, 24 of 24)

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- [Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling](https://mlsanity.com/paper/2410.13645/) (2024)
- [Learning Memory and Material Dependent Constitutive Laws](https://mlsanity.com/paper/2502.05463/) (2025)
- [Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches](https://mlsanity.com/paper/2503.00575/) (2025)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)](https://mlsanity.com/paper/2604.07746/) (2026)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [A machine learning-based viscoelastic-viscoplastic model for epoxy nanocomposites with moisture content](https://mlsanity.com/paper/2305.08102/) (2023)
- [Differentiable neural network representation of multi-well, locally-convex potentials](https://mlsanity.com/paper/2506.17242/) (2025)
- [Equivariant graph convolutional neural networks for the representation of homogenized anisotropic microstructural mechanical response](https://mlsanity.com/paper/2404.17584/) (2024)
- [Fully data-driven inverse hyperelasticity with hyper-network neural ODE fields](https://mlsanity.com/paper/2506.08146/) (2025)
- [Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models](https://mlsanity.com/paper/2601.03367/) (2026)
- [Physics-Informed Discovery of Yield Functions in Plasticity via Convex Neural Representations](https://mlsanity.com/paper/2606.19375/) (2026)
- [Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling](https://mlsanity.com/paper/2607.20339/) (2026)

## Abstract (arXiv)

Data-driven constitutive modeling is an emerging field in computational solid mechanics with the prospect of significantly relieving the computational costs of hierarchical computational methods. Traditionally, these surrogates have been trained using datasets which map strain inputs to stress outputs directly. Data-driven constitutive models for elastic and inelastic materials have commonly been developed based on artificial neural networks (ANNs), which recently enabled the incorporation of physical laws in the construction of these models. However, ANNs do not offer convergence guarantees and are reliant on user-specified parameters. In contrast to ANNs, Gaussian process regression (GPR) is based on nonparametric modeling principles as well as on fundamental statistical knowledge and hence allows for strict convergence guarantees. GPR however has the major disadvantage that it scales poorly as datasets get large. In this work we present a physics-informed data-driven constitutive modeling approach for isostropic and anisotropic materials based on probabilistic machine learning that can be used in the big data context. The trained GPR surrogates are able to respect physical principles such as material frame indifference, material symmetry, thermodynamic consistency, stress-free undeformed configuration, and the local balance of angular momentum. Furthermore, this paper presents the first sampling approach that directly generates space-filling points in the invariant space corresponding to bounded domain of the gradient deformation tensor. Overall, the presented approach is tested on synthetic data from isotropic and anisotropic constitutive laws and shows surprising accuracy even far beyond the limits of the training domain, indicating that the resulting surrogates can efficiently generalize as they incorporate knowledge about the underlying physics.