Physics-constrained symbolic model discovery for polyconvex incompressible hyperelastic materials

TL;DR

The paper addresses learning physically admissible hyperelastic energy functionals for incompressible, isotropic materials from sparse data. It introduces PNAM, a polyconvex neural additive model where each shape function $\psi_i(I_i)$ is realized by an ICNN ensuring convexity in $I_i$, enabling polyconvexity of the energy expressed as $\psi(I_1,I_2)=\psi_1(I_1)+\psi_2(I_2)$ with an incompressibility penalty $-\tfrac{1}{2}p(I_3-1)$ and a counterbalance term $-\alpha_0(I_3-1)$ plus a zero-energy baseline. A second SR stage converts each univariate basis into compact symbolic expressions, yielding interpretable multivariate energy functionals with guaranteed polyconvexity and reduced runtime. Numerical tests on Treloar data and a synthetic composite show competitive accuracy with symbolic forms that generalize well beyond the training regime, while providing greater interpretability and computational efficiency for large-scale simulations.

Abstract

We present a machine learning framework capable of consistently inferring mathematical expressions of hyperelastic energy functionals for incompressible materials from sparse experimental data and physical laws. To achieve this goal, we propose a polyconvex neural additive model (PNAM) that enables us to express the hyperelastic model in a learnable feature space while enforcing polyconvexity. An upshot of this feature space obtained via the PNAM is that (1) it is spanned by a set of univariate basis that can be re-parametrized with a more complex mathematical form, and (2) the resultant elasticity model is guaranteed to fulfill the polyconvexity, which ensures that the acoustic tensor remains elliptic for any deformation. To further improve the interpretability, we use genetic programming to convert each univariate basis into a compact mathematical expression. The resultant multi-variable mathematical models obtained from this proposed framework are not only more interpretable but are also proven to fulfill physical laws. By controlling the compactness of the learned symbolic form, the machine learning-generated mathematical model also requires fewer arithmetic operations than its deep neural network counterparts during deployment. This latter attribute is crucial for scaling large-scale simulations where the constitutive responses of every integration point must be updated within each incremental time step. We compare our proposed model discovery framework against other state-of-the-art alternatives to assess the robustness and efficiency of the training algorithms and examine the trade-off between interpretability, accuracy, and precision of the learned symbolic hyperelastic models obtained from different approaches. Our numerical results suggest that our approach extrapolates well outside the training data regime due to the precise incorporation of physics-based knowledge.