Convex Equilibrium Potentials for Monge--Amp\`ere: Curvature Bounds and A Posteriori Evaluation

Abstract

Equilibrium neural networks define their hidden state as the fixed point of a contraction, so their effective depth is set at evaluation time rather than by the architecture. Using them for fully nonlinear PDEs raises two issues that explicit networks avoid: second derivatives may grow with the effective depth, and the potential is only ever evaluated approximately. We study nonnegative softplus equilibrium networks with contraction factor $q<1$, a convex class suited to the Monge--Amp\`ere equation. A positivity cancellation in the resolvent shows that the Hessian grows at most like $(1-q)^{-2}$, rather than the $(1-q)^{-3}$ of a direct estimate, and an explicit one-neuron family shows that this exponent is sharp; third derivatives satisfy a sharp $(1-q)^{-3}$ bound. Residual-based a posteriori estimates then control the state, its input derivatives and, in two dimensions, the Hessian determinant at an approximate equilibrium. Combined with interval arithmetic on complete covers of the domain, they yield rigorous uniform bounds on the Monge--Amp\`ere residual and, through a comparison argument, on the solution error of a fixed trained network. On two manufactured Dirichlet problems on the unit disk, all six trained models are certified with residual below $0.25$ and boundary error below $0.01$. Adaptive stopping halves float64 evaluation time at best, whereas certified adaptive evaluation remains costlier than fixed depth.