AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-23
Convex Equilibrium Potentials for Monge--Amp\`ere: Curvature Bounds and A Posteriori Evaluation
Equilibrium neural networks define their state as a fixed point, which raises second-derivative growth and approximate evaluation problems for fully nonlinear PDEs such as Monge–Ampère. The authors show that for a convex class of such networks the Hessian grows at most like (1−q)^−2 rather than (1−q)^−3, and that this exponent is sharp; they combine residual-based a posteriori estimates with interval arithmetic. On two manufactured Dirichlet problems on the unit disk, all six trained models were certified with residual below 0.25 and boundary error below 0.01, while adaptive stopping halves float64 evaluation time at best and certified adaptive evaluation costs more than fixed depth.