AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-26
Universal neural network approximation of symmetric polyconvex energies and envelopes
Symmetric polyconvexity gives a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for relaxing non-convex energies. The authors show that the standard weight restrictions on ReLU input-convex networks do not allow universal approximation of the required representers, and they introduce symmetric polyconvex neural networks (SPCNNs). They prove a universal approximation theorem for symmetric polyconvex energies and illustrate the method on stress-based regression and on the envelope of a double-well potential.