AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-26
Implicit Neural Representations for Variational Problems on Graphons
The authors study variational problems over graphons, which arise in extremal graph theory and in the large deviation theory of dense random graphs. They represent graphons with implicit neural networks and optimise them by gradient descent. The framework rediscovered known optimal graphons for generalised Turán problems without human intervention, and for open cases it produced candidates, including a previously unreported family of extremal structures. For the triangle upper tail, the candidates improve on the best-known construction of Lubetzky and Zhao, and the structures led to new theorems.