Convex Equilibrium Potentials for Monge--Ampère: 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--Ampère 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--Ampère 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.
Related
- A counterexample to quasiconcavity for the Gaussian p-Capacity potentials in convex rings
- Muon meets Tamed Langevin: Momentum Preconditioning beyond Convex and gradient-Lipschitz Potentials
- A remark on the compactness of manifolds with non-negative Ricci curvature and uniformly convex boundary
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
- A Bilevel Market Equilibrium Model for Natural Gas Pipeline Network Operations: Convex Relaxations and Single-level Reformulations
BibTeX
@misc{vazquez2026convex,
title = {{Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation}},
author = {Edison Vázquez and Yingdong Lu and Tomasz Nowicki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31386},
eprint = {2610.31386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31386},
}
RIS
TY - UNPB
TI - Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation
AU - Edison Vázquez
AU - Yingdong Lu
AU - Tomasz Nowicki
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.31386
UR - https://arxiv.org/abs/2610.31386
AB - 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--Ampère 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--Ampère 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.
ER -