# Model-data-driven constitutive responses: application to a multiscale computational framework

arXiv: https://arxiv.org/abs/2104.02650 · HTML: https://mlsanity.com/paper/2104.02650/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 54 references that are in the map.
- First posted on arXiv 6 Apr 2021.

## Builds on (most cited first, 7 of 7)

- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials](https://mlsanity.com/paper/1807.09829/) (2018)
- [Model-Free Data-Driven Inelasticity](https://mlsanity.com/paper/1808.10859/) (2018)
- [A machine learning based plasticity model using proper orthogonal decomposition](https://mlsanity.com/paper/2001.03438/) (2020)
- [Gradient-enhanced kriging for high-dimensional problems](https://mlsanity.com/paper/1708.02663/) (2017)
- [Exploring the 3D architectures of deep material network in data-driven multiscale mechanics](https://mlsanity.com/paper/1901.04832/) (2019)
- [On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning](https://mlsanity.com/paper/2007.07749/) (2020)

## Built on by (most cited first, 6 of 6)

- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [Interval and fuzzy physics-informed neural networks for uncertain fields](https://mlsanity.com/paper/2106.13727/) (2021)
- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)

## Abstract (arXiv)

Computational multiscale methods for analyzing and deriving constitutive responses have been used as a tool in engineering problems because of their ability to combine information at different length scales. However, their application in a nonlinear framework can be limited by high computational costs, numerical difficulties, and/or inaccuracies. In this paper, a hybrid methodology is presented which combines classical constitutive laws (model-based), a data-driven correction component, and computational multiscale approaches. A model-based material representation is locally improved with data from lower scales obtained by means of a nonlinear numerical homogenization procedure leading to a model-data-driven approach. Therefore, macroscale simulations explicitly incorporate the true microscale response, maintaining the same level of accuracy that would be obtained with online micro-macro simulations but with a computational cost comparable to classical model-driven approaches. In the proposed approach, both model and data play a fundamental role allowing for the synergistic integration between a physics-based response and a machine learning black-box. Numerical applications are implemented in two dimensions for different tests investigating both material and structural responses in large deformation.