Certified neural approximations for parabolic obstacle problems

Davood Damircheli

math.NAOct 16, 2026

Certified neural approximations for parabolic obstacle problems

Authors: Davood Damircheli

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

Abstract

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.

Figures & tables

	Quantity	Status
(a)	Theorem 3.2 , its corollaries, Appendix A	Analytic inequalities between continuum integrals.
(b)	Per-term upper sums and the absolute bounds $\widehat{F}_{\rm dir},\widehat{G}_{\rm dir},\widehat{G}_{\rm AR},\widehat{E}_{\rm dir}$	Outward-rounded evaluation of (a) by Lemma 5.2 ; valid under Sections B.1 and B.2 . Printed as decimal ceilings of full-precision endpoints, recomputed in $192$ -bit Arb.
(c)	Relative bounds $\widehat{F}_{\rm dir}/N$ , $\widehat{G}_{\rm dir}/N$ , $\widehat{G}_{\rm AR}/N$ ; $N_{\rm lo}$ and ratios over it	Numerator from (b); denominator a $192$ -bit lower endpoint of the closed-form $N$ ( Section B.3 ) or $N_{\rm lo}$ from downward-rounded lower sums, printed as a decimal floor.
(d)	Errors $E(v)$ , $E(\widehat{v})$ (weak norm ( 9 )), flux and gradient errors, float64 integrals, slacks, shares, effectivities, slopes, substitutions, the finite-difference reference, all field and partition figures	Diagnostics (float64 quadrature, sampling or a reference solve); no certified status.
(e)	Implementation checks ( Section B.2 )	Tests of the implementation; not a proof of enclosure validity.

Table 1: Status of the reported quantities.

Seed	$E(\widehat{v})$	$\widehat{F}_{\rm dir}$	$\widehat{F}_{\rm dir}/N$	$\widehat{G}_{\rm dir}/N$	$\widehat{G}_{\rm AR}/N$	$\widehat{E}_{\rm dir}$	eff.
0	0.0847	0.06847	0.260	0.330	0.319	0.1040	1.23
1	0.0866	0.06923	0.263	0.336	0.324	0.1070	1.24
2	0.0834	0.06723	0.256	0.325	0.314	0.1031	1.24

Table 2: Stefan, uniform

512^{2}

partition. Validated:

\widehat{F}_{\rm dir}

, the ratios and

\widehat{E}_{\rm dir}

. Diagnostic:

E(\widehat{v})

and the effectivity

\widehat{E}_{\rm dir}/E(\widehat{v})

Figure 1: Stefan, relative bounds on uniform partitions: unsplit direct flux bound ( 30 ) (solid), estimator-route projected bound

\widehat{G}_{U}

of ( 45 ) (dashed), and the split direct bounds at

512^{2}

cells (stars:

\widehat{F}_{\rm dir}

; crosses:

\widehat{G}_{\rm dir}

). Dotted: ratio

1

. All values validated.

Prot.	Seed	$\widehat{F}_{\rm dir}$	$\widehat{F}_{\rm dir}/N$	$\widehat{G}_{\rm dir}/N$	$\widehat{G}_{\rm AR}/N$	$\widehat{U}_{R}$	$\widehat{U}_{S}$	$U_{c}$	h
A	0	0.2718	0.395	0.643	0.524	0.5735	0.02767	0.008565	1.9
A	1	0.3208	0.466	0.697	0.579	0.4994	0.03391	0.02181	1.9
A	2	0.3188	0.463	0.731	0.596	0.6724	0.03394	0.01678	1.9
B	0	0.1936	0.282	0.419	0.364	0.1777	0.01803	0.005216	4.7
B	1	0.2331	0.339	0.478	0.415	0.1798	0.02246	0.01136	4.6
B	2	0.2263	0.329	0.482	0.414	0.2182	0.02169	0.009223	4.6

Table 3: Long-trained expanding disk under the two refinement protocols of Section 6.1 . Validated: all bounds, ratios and per-term sums.

\widehat{G}_{\rm dir}/N

is the protocol target. Wall time in hours.

Schedule	Seed	$E(\widehat{v})$	$\widehat{G}_{U}/N$	$\widehat{F}_{\rm dir}^{\rm ns}/N$	$\widehat{G}_{\rm dir}^{\rm ns}/N$	$\widehat{F}_{\rm dir}/N$	$\widehat{G}_{\rm dir}/N$	$\widehat{G}_{\rm AR}/N$
Baseline	0	0.2415	7.26	2.97	3.01	0.704	1.17	0.974
Baseline	1	0.2677	7.29	2.98	3.02	0.719	1.19	1.01
Baseline	2	0.2460	8.24	3.37	3.41	0.760	1.29	1.06
Long	0	0.0917	9.37	3.85	3.90	1.02	1.62	1.32
Long	1	0.1021	9.07	3.74	3.79	1.11	1.68	1.39
Long	2	0.1040	10.1	4.15	4.20	1.12	1.77	1.43

Table 4: Expanding disk, frozen adaptive

10^{6}

-cell partitions: certificates of the projected output on the same enclosures.

\widehat{G}_{U}

: estimator route ( 45 ); superscript ns: unsplit direct bounds ( 30 ); last three columns: split direct bounds. Ratios validated;

E(\widehat{v})

diagnostic.

Partition	Cells	$\frac{\widehat{F}_{\rm dir}}{N}$	$\frac{\widehat{G}_{\rm dir}}{N}$	$\frac{\widehat{G}_{\rm AR}}{N}$	$\mathcal{L}$	$\widehat{U}_{S}$	$2U_{c}$	slack $\widehat{U}_{R}$ , $\widehat{U}_{S}$ , $U_{c}$
adaptive	99996	6.39	11.5	8.73	0.64	0.30	0.06	139192	790.2	31736
adaptive	1000000	1.02	1.62	1.32	0.35	0.34	0.31	1925	22.6	3870
uniform	2097152	0.514	0.897	0.687	0.56	0.28	0.16	788	4.8	517
A	3499994	0.395	0.643	0.524	0.39	0.37	0.23	330	3.8	434
B	4500000	0.359	0.562	0.471	0.32	0.42	0.26	221	3.5	400
B	5999995	0.318	0.483	0.411	0.27	0.43	0.30	147	2.8	362

Table 5: Long disk, seed 0: validated direct ratios, shares of

\widehat{F}_{\rm dir}^{2}

carried by

\mathcal{L}

\widehat{U}_{S}

and

2U_{c}

, and the slack of each per-term upper sum against the float64 integral of the same density on the same partition (diagnostic). The first two rows are adaptive partitions of

10^{5}

and

10^{6}

cells; the others follow Protocols A and B.

Seed	Partition	Cells	$\widehat{F}_{\rm dir}$	$\widehat{F}_{\rm dir}/N$	$\widehat{G}_{\rm dir}/N$	$\widehat{G}_{\rm AR}/N$
0	adaptive	99996	2.094	6.65	11.2	9.37
	uniform $128^{3}$	2097152	0.2591	0.822	1.22	1.12
	A	3499994	0.1888	0.599	0.891	0.814
	B	4500000	0.1766	0.561	0.794	0.742
	B	5999995	0.1667	0.529	0.732	0.691
	B	8000000	0.1583	0.503	0.681	0.648

Table 6: Expanding ellipse, direct bounds of the projected output (validated). Diagnostic weak-norm errors

E(\widehat{v})

0.1186

0.1183

0.1238

for seeds

0

1

2

. Protocol B is applied to seed 0 only, since its target is not met there.

Run	Flux	Multiplier	$U_{R}$	$U_{S}$	$U_{S}$ on ring	$F/N$	$G_{\rm AR}/N$
S0	$\sigma_{\phi}$	$\lambda_{\psi}$	$1.08\cdot 10^{-3}$	$1.30\cdot 10^{-2}$	$1.24\cdot 10^{-2}$	0.363	0.386
S1	$\sigma_{\phi}$	$\lambda$ (exact)	$4.66\cdot 10^{-1}$	$1.30\cdot 10^{-2}$	$1.24\cdot 10^{-2}$	0.608	0.850
S2	$\sigma_{\phi}$	$\mu^{\ast}$	$4.83\cdot 10^{-4}$	$1.30\cdot 10^{-2}$	$1.24\cdot 10^{-2}$	0.363	0.378
S3	$\sigma$ (exact)	$\lambda_{\psi}$	$4.69\cdot 10^{-1}$	$8.71\cdot 10^{-3}$	$7.04\cdot 10^{-3}$	0.572	0.785
S4	$\sigma$ (exact)	$\lambda$ (exact)	$1.22\cdot 10^{-4}$	$8.71\cdot 10^{-3}$	$7.04\cdot 10^{-3}$	0.297	0.304
S5	$\sigma$ (exact)	$\mu^{\ast}$	$3.54\cdot 10^{-5}$	$8.71\cdot 10^{-3}$	$7.04\cdot 10^{-3}$	0.297	0.301

Table 7: Expanding ellipse, seed 0, final Protocol B partition: float64 integrals of the per-term densities (no enclosure; diagnostic) for the trained projected state with the trained or substituted flux and multiplier. “Ring”: the part of

\{v<g\}

outside the exact contact set. The Protocol A partition gives the same values to four digits.

Panel of Figure 2 — Figure 2: Seed- $0$ fields at $t=0.25,0.5,0.75$ on a $401\times 401$ grid (diagnostic, float64). Top two rows, long disk: projected state error $|u-\widehat{v}|$ , $\widehat{v}=\max\{u_{\theta},g\}$ , and multiplier error $|\lambda-\lambda_{\psi}|$ ; dashed: exact free boundary $r=R(t)$ . Bottom two rows, ellipse: $|u-\widehat{v}|$ , large on the ring between the exact contact set (dashed) and the edge of $\{u_{\theta}<g\}$ , and the trained multiplier $\lambda_{\psi}$ on the colour scale of the exact $\lambda$ , numerically zero.

Figure 3: Ellipse, seed $0$ : the final Protocol B partition ( $8\cdot 10^{6}$ space–time cells) cut at $t=0.25,0.5,0.75$ ; a cell is drawn if $t_{0}\leq t<t_{1}$ . Top: refinement depth $-\log_{2}(\text{cell side})$ . Bottom: the Protocol B marking density $\log_{10}(\mu_{K}/|K|)$ . The marking concentrates on the ring between the exact contact set (dashed; not used by the partition) and the edge of the region where the projection is active.

		validated						diagnostic
Cells	Seed	$\widehat{F}_{\rm dir}$	$\widehat{G}_{\rm AR}$	$N_{\rm lo}$	$\frac{\widehat{F}_{\rm dir}}{N_{\rm lo}}$	$\frac{\widehat{G}_{\rm dir}}{N_{\rm lo}}$	$\frac{\widehat{G}_{\rm AR}}{N_{\rm lo}}$	flux	grad.	$\frac{N_{\rm lo}}{N_{\rm ref}}$
$1024^{2}$	0	1.445	1.629	0.211	6.85	7.78	7.72	1.247	1.051	0.244
$1024^{2}$	1	1.447	1.634	0.206	7.01	7.98	7.91	1.245	1.057	0.239
$1024^{2}$	2	1.440	1.621	0.217	6.62	7.52	7.46	1.250	1.041	0.252
$512^{2}$	0	1.452	1.637	0.201	7.19	8.17	8.11	1.247	1.051	0.234
$512^{2}$	1	1.454	1.642	0.197	7.36	8.38	8.31	1.245	1.057	0.228

Table 8: Rising obstacle (no closed-form solution). Validated:

\widehat{F}_{\rm dir}

\widehat{G}_{\rm AR}

(ceilings),

N_{\rm lo}

(floor) and the ratios (ceilings). Diagnostic, against the reference (

N_{\rm ref}=0.86486\pm 1.1\cdot 10^{-4}

): relative flux and projected-gradient errors and

N_{\rm lo}/N_{\rm ref}

Figure 4: Rising obstacle, seed $0$ (diagnostic; reference on $4096^{2}$ ). (a) Reference solution; solid: boundary of the reference contact set; dashed: boundary of $\{v\leq g\}$ for the network. (b) Error of the projected gradient. (c) Constitutive residual density $|\widehat{S}|^{2}$ , large on all of $\{v\leq g\}$ , where $\widehat{v}=g$ while the flux follows the unconstrained state.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Primitive	Enclosure argument	Assumptions
$+,-,\times,\div,\sqrt{\ }$ (binary64)	Correctly rounded IEEE 754 operations; each endpoint moved one unit in the last place outward after every operation.	Round to nearest; no fused multiply–add or extended intermediates; no flush to zero.
Sums of products (TM1 path)	Sequences of the operations above, each rounded outward, so the result encloses the exact sum in any order; no BLAS.	As above.
$\tanh$ and derivatives	Monotonicity; endpoint values from an Arb-based dyadic lookup with Taylor remainder bounds, converted outward; derivative ranges from $P_{q}(\tanh)$ ; no system transcendental as endpoint oracle.	Ball semantics of Arb [ 38 ] .
Taylor-model remainders	Lagrange remainders with second-derivative bounds from the previous row on the enclosed pre-activation range $I$ , with radius $\sup_{a\in I}\|a-a_{0}\|$ ; the expansion point $a_{0}$ is checked to lie in $I$ .	Rows above.
Data $f,g,u_{0}$ and derivatives	Arb ball evaluation on the cell at $\geq 64$ bits, converted outward.	Ball semantics of Arb.
Volumes, reductions, constants	Exact dyadic volumes; endpoints converted exactly to binary rationals; Arb ball sums; $\overline{c}_{F}^{\,2}$ an upward-rounded rational; $N$ a $192$ -bit lower endpoint.	Ball semantics of Arb.

Appendix

Table 9: Primitives of the finite evaluation, the argument for their outward enclosure, and its assumptions.

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