Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation

Edison Vázquez; Yingdong Lu; Tomasz Nowicki

math.NAOct 22, 2026

Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation

Authors: Edison Vázquez, Yingdong Lu, Tomasz Nowicki

Organizations: Departamento de Matemáticas, CINVESTAV, México · IBM Research, Thomas J. Watson Research Center, Yorktown Heights, New York, USA

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.

Figures & tables

Figure 1: Analytic sharpness family from Theorem 4.4 . Left: exact curvature and the two upper bounds for

A=w=1

. Right: the rescaled curvature tends to

1/8

as the contraction gap decreases. Curves are evaluations of the displayed closed forms.

Quantity	Maximum ratio	Violations
State error	0.98924	0
First derivative error	0.77929	0
Second derivative error	0.69098	0
Hessian error	0.17737	0
Determinant error	0.07730	0
Curvature bound	0.20601	0

Table 1: Independent float64 consistency experiment, 120 networks. Error rows show the largest observed error-to-bound ratio, maximizing over coordinates where applicable. The last row is

\max_{r}(S\gamma)_{r}

, whose bound is one. A violation means a ratio exceeding

1+10^{-8}

. These are numerical checks against approximate reference quantities.

Item	Executed configuration
Adam	1,200 steps; initial learning rate 0.02; factor 0.35 at steps 780 and 1,080; fresh 128 interior and 64 boundary points per step
L-BFGS	Up to 100 iterations, strong-Wolfe line search; fixed 512 interior and 128 boundary points
Evaluation	2,048 interior points and 512 boundary points per model
Depth sweep	$L\in\{4,8,12,16,24,32\}$ on the same 2,048 interior points
Float64 timing	512 points/model; five repetitions; tolerances $10^{-4}$ and $10^{-6}$
Interval timing	Separate eight-point/model cohort; five repetitions; same tolerances

Table 2: Protocol of the completed six-model run.

Model	Rel. $\ell^{2}$ (%)	Max. $\|u-u^{*}\|$	Max. $\|\det H-f\|$	Max. boundary	$\left\lVert W\right\rVert_{\infty}$
P1/0	0.2478	0.003927	0.045998	0.004150	0.6584
P1/1	0.2937	0.004282	0.054399	0.004573	0.6135
P1/2	0.3125	0.004958	0.052480	0.005022	0.5817
P2/0	0.2808	0.003626	0.102253	0.003961	0.6466
P2/1	0.2872	0.003799	0.080306	0.003872	0.6580
P2/2	0.3144	0.005618	0.089144	0.005954	0.6330

Table 3: Float64 sample metrics. A model label P1/0 means problem P1, seed 0. Interior maxima use 2,048 points; boundary maxima use 512 points. Values are rounded for display and are not interval bounds.

Figure 2: Determinant evaluation error over iteration depth for all six models. Each curve is the maximum over the same 2,048 evaluation points. Blue: float64 estimator; orange: outward-rounded Arb upper bound; green: error against the float64 reference. The gray line marks

10^{-11}

, the additive diagnostic floor; the reference diagnostic is added separately in violation counting. Curves near this scale should not be interpreted as exact-error measurements.

Figure 3: Matched-cohort evaluation timings. Ratios above one favor adaptive evaluation; ratios below one favor fixed 32 iterations. Left: float64 estimator on 512 points/model, compared with fixed 32 without certification. Right: full Arb stopping on eight points/model, compared with fixed 32 plus Arb certification. Both comparisons include the adaptive certificate cost and use five repetitions. The cohorts and baselines differ; their throughputs must not be compared directly.

Figure 4: Additional P2, seed-2 certification, without retraining. The residual upper bound decreases from

0.437694

0.254972

at depth 8 and then to

0.249951

at depth 9 (displayed bounds rounded up). The cover remains complete throughout. Stage 1 computes 1,850 new certificates and stage 2 computes 10, retaining 15,937 boxes at completion.

Model	$R$	$B_{\partial}$	Direct $U$	Barrier $U$	Boxes
P1/0	0.232852	0.010000	0.107237	0.251273	3,332
P1/1	0.243891	0.009989	0.114571	0.256915	3,332
P1/2	0.217254	0.009996	0.110518	0.243048	3,332
P2/0	0.249993	0.009997	0.053797	0.259994	11,310
P2/1	0.249997	0.009999	0.065538	0.259997	11,668
P2/2	0.249951	0.009989	0.046754	0.259965	15,937

Table 4: Uniform upper bounds on the closed disk and its boundary. P2/2 uses the additional refinement; all other rows use the original covers. Direct

U

bounds

|u_{\theta}-u^{*}|

using the known manufactured solution. Barrier

U=B_{\partial}+\sqrt{R}/2

uses Proposition 6.3 . Every displayed bound is rounded upward to six decimal places.

Figure 5: Samples and uniform bounds have different meanings. Left: observed PDE residual maxima versus interval upper bounds over the disk. Right: observed solution-error maxima, direct uniform bounds using the manufactured target, and the explicit barrier bound. P2/2 includes the final refinement. Curves connect model labels for readability; no trend across seeds is assumed.

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Certified neural approximations for parabolic obstacle problems

Davood Damircheli

Department of Mathematics, Louisiana State University, Baton Rouge, 70803, LA, USA

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Residual-trained neural surrogates carry no error guarantee, a limitation for free-boundary problems such as phase change and contact. We derive guaranteed error bounds for neural approximations of parabolic obstacle problems in first-order state--flux--multiplier form. The main result bounds the flux error of the network and the energy error of its feasible projection

\max\{v,g\}

by the constitutive residual, the initial mismatch, the complementarity defect, and the PDE residual weighted by

c_F^2/α

, without the coercivity and norm-equivalence constants of the first-order formulation. It extends the Apushkinskaya--Repin majorant for the heat operator to general coefficients, free multipliers and a flux bound. A penetration-consistency result shows that neural triples with arbitrarily small bound exist. Outward-rounded cellwise Taylor-model enclosures turn the bound into a finite computation valid on any adaptive partition. The flux and gradient errors of the projected output are certified below

0.27

and

0.33

of the solution scale for a one-dimensional Stefan problem and, after adaptive refinement, below one half for all three trained networks on a two-dimensional expanding disk. On an expanding ellipse the gradient bound stays above one half; substitution diagnostics trace the floor of the trained networks to the projected state near the free boundary, and the remaining gap to enclosure overhead. Without a closed-form solution, a validated lower bound replaces the solution norm; there the enclosure is nearly exact, but the trained networks have errors above the solution scale and no useful accuracy is certified. The bounds control a state--flux norm, not the multiplier or the free-boundary location.

Abstract

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