Benchmarking physics-informed frameworks for data-driven hyperelasticity

Data-driven methods have changed the way we understand and model materials. However, while providing unmatched flexibility, these methods have limitations such as reduced capacity to extrapolate, overfitting, and violation of physics constraints. Recently, frameworks that automatically satisfy these requirements have been proposed. Here we review, extend, and compare three promising data-driven methods: Constitutive Artificial Neural Networks (CANN), Input Convex Neural Networks (ICNN), and Neural Ordinary Differential Equations (NODE). Our formulation expands the strain energy potentials in terms of sums of convex non-decreasing functions of invariants and linear combinations of these. The expansion of the energy is shared across all three methods and guarantees the automatic satisfaction of objectivity, material symmetries, and polyconvexity, essential within the context of hyperelasticity. To benchmark the methods, we train them against rubber and skin stress-strain data. All three approaches capture the data almost perfectly, without overfitting, and have some capacity to extrapolate. This is in contrast to unconstrained neural networks which fail to make physically meaningful predictions outside the training range. Interestingly, the methods find different energy functions even though the prediction on the stress data is nearly identical. The most notable differences are observed in the second derivatives, which could impact performance of numerical solvers. On the rich data used in these benchmarks, the models show the anticipated trade-off between number of parameters and accuracy. Overall, CANN, ICNN and NODE retain the flexibility and accuracy of other data-driven methods without compromising on the physics. These methods are ideal options to model arbitrary hyperelastic material behavior.

Automated summary

Data-driven methods have revolutionized the understanding and modeling of materials by providing unprecedented flexibility. However, they also have limitations such as reduced extrapolation capacity and violation of physics constraints. In this review, we compare and extend three promising data-driven methods - CANN, ICNN, and NODE - that automatically satisfy these requirements within the context of hyperelasticity. By training them against stress-strain data, we find that all three methods capture the data perfectly without overfitting and exhibit some extrapolation capability. The models show a trade-off between number of parameters and accuracy, but retain flexibility and accuracy without compromising on the physics.