Universal neural network approximation of symmetric polyconvex energies and envelopes

Approximating the effective anisotropic hyperelastic response of a representative volume element (RVE) requires constitutive expressiveness, mechanical consistency, and computational efficiency. This work develops a polyconvex physics-augmented neural network (PANN) construction based on paired directional features whose orientations, stretch powers, and determinant exponents adapt to the RVE response. The features are composed with convex, componentwise nondecreasing cores, realized as an input-convex neural network (ICNN) or an input-convex Kolmogorov-Arnold network (ICKAN). The complete energies satisfy objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth conditions by construction, and nonnegative energy is certified separately for the selected models. On a multicavity and a single-cavity periodic cell, these energies are compared with an Unconstrained energy, trained identically without the curvature and growth constraints, and with three hyperreduced projection-based reduced-order models (HPROMs), which serve as physics-based references. On the first cell, learning the features lowers the stress error sevenfold, and six features suffice on both. The constrained energies remain below one percent in stress error and in the displacement error of a tensile-coupon FE$^2$ application with a full-order reference, whereas the Unconstrained energy and the HPROMs are more accurate inside the data. Beyond the data, however, the Unconstrained energy loses rank-one convexity, and under confined compression it stops below half of the 500 MPa that the constrained energies reach. These results quantify what the guarantees cost in accuracy and what they secure beyond the data.

We establish two related but logically distinct results for shallow ReLU$^k$ neural networks on the unit sphere $\SS^d$. First, for an arbitrary set of inner neural-network parameters, the best $\mathcal{L}^2(\SS^d)$ approximation of a fixed target function with smoothness $r>\tfrac{d+2k+1}{2}$ admits an asymptotic lower bound given by a constant multiple of $n^{-1/2}\underline{h}^{k+1/2}$, where $\underline{h}$ denotes the antipodal separation distance of the normalized inner-parameter set. This lower bound depends explicitly on the parameter configuration through $\underline{h}$ and applies without additional assumptions on the parameters. Second, for antipodally quasi-uniform parameters, $\underline{h}\simeq n^{-1/d}$, and the lower bound establishes the exact saturation order $n^{-\frac{d+2k+1}{2d}}$ for such parameter families: a target function with regularity greater than $\frac{d+2k+1}{2}$ and satisfying the required parity condition can be approximated at this rate, whereas approximation at any strictly faster rate forces the target function to be zero. Our results therefore place linearized neural-network approximation within the classical saturation framework and show that, although ReLU$^k$ network spaces can outperform finite elements of the same degree, this advantage is intrinsically limited.

We present a physics-based neural network framework for the discovery of constitutive models in fully coupled thermomechanics. In contrast to classical formulations based on the Helmholtz energy, we adopt the internal energy and a dissipation potential as primary constitutive functions, expressed in terms of deformation and entropy. This choice avoids the need to enforce mixed convexity--concavity conditions and facilitates a consistent incorporation of thermodynamic principles. In this contribution, we focus on materials without preferred directions or internal variables. While the formulation is posed in terms of entropy, the temperature is treated as the independent observable, and the entropy is inferred internally through the constitutive relation, enabling thermodynamically consistent modeling without requiring entropy data. Thermodynamic admissibility of the networks is guaranteed by construction. The internal energy and dissipation potential are represented by input convex neural networks, ensuring convexity and compliance with the second law. Objectivity, material symmetry, and normalization are embedded directly into the architecture through invariant-based representations and zero-anchored formulations. We demonstrate the performance of the proposed framework on synthetic and experimental datasets, including purely thermal problems and fully coupled thermomechanical responses of soft tissues and filled rubbers. The results show that the learned models accurately capture the underlying constitutive behavior. All code, data, and trained models are made publicly available via https://doi.org/10.5281/zenodo.19248596.