Automated material model discovery | EUCLID

Learning stress-strain models without(!) stress data 

Despite the recent advances in data-driven methods from neural networks to Gaussian processes, constitutive modelling of materials remains embedded in a supervised setting where the stress-strain pairs are assumed to be available. However, in most common experimental setups, it is difficult to probe the entire stress-strain space, while getting such labelled data is expensive via multiscale simulations. The biggest challenge is – how does one even measure full stress tensors (forces are only boundary-averaged projections of stress tensors) for learning the stress-strain relations? 

To bypass these challenges, we proposed a new data-driven framework called EUCLID which stands for – Efficient Unsupervised Constitutive Law Identification and Discovery ( 

The approach is unsupervised, i.e., it requires no stress data but only displacement and global force data, which are realistically available through mechanical testing and digital image correlation (DIC) techniques. In contrast to supervised learning based on stress labels, the problem of unsupervised discovery is solved by leveraging physical laws such as conservation of linear momentum in the bulk and at the loaded boundary of a test specimen. EUCLID enables discovering physically consistent models embodied by either:

We demonstrated several benchmarks on the discovery of hyperelastic and elastoplastic constitutive models without using any or limited stress data. Together with several collaborators across different communities, we are currently exploring different applications of EUCLID — from soft biological tissues to metamaterials.


All codes and data are aggregated here:



TU Delft, Mekelweg 2, Delft 2628 CD, Netherlands