Compression of Polyconvex Envelopes of Isotropic Functions via Monotonic Input Convex Neural Networks

ABSTRACT This work presents a novel neural‐network compression approach for polyconvex envelopes of isotropic functions. The approach relies on a classical sufficient criterion for polyconvexity and is particularly suited for the representation of determinant‐constrained energy densities arising in non‐linear elasticity. Compared with existing compression methods based on the necessary and sufficient characterisation of polyconvex isotropic functions, the proposed framework reduces computational costs due to the domain reduction through the restriction to the positive octant in the signed singular value space. The underlying neural‐network architecture employs input‐convex neural networks (ICNNs) with non‐negative weight constraints to enforce the required convexity and monotonicity properties. The additional symmetry and inequality conditions characterising the polyconvex envelope are incorporated weakly through the loss function during training. Although the employed criterion is only sufficient and thus generally yields only a lower bound on the polyconvex envelope, numerical experiments based on the classical Saint Venant–Kirchhoff energy demonstrate that the proposed approach produces accurate approximations in practice while offering a computationally more efficient alternative to existing methods.