SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization

Wondesen Teshome Bekele; Marco D'Oria

cs.LGSep 27, 2026

SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization

Authors: Wondesen Teshome Bekele, Marco D'Oria

Organizations: Department of Engineering and Architecture, University of Parma, Parma, Italy · University School for Advanced Studies IUSS Pavia, Pavia, Italy

Abstract

Many probabilistic predictors must satisfy exact structure in every stochastic realization, yet common hard-constraint approaches form predictions in ambient coordinates and then correct or project them. We introduce SLP-ProbHard, a cross-family, representation-centered framework for probabilistic hard-constrained learning when explicit structural parameterizations are available. Its core object, a Structural Feasible Latent Parameterization (SFLP), combines a structural latent law $Z \sim P^Z_θ(\cdot\mid x)$ with a feasible map $Y=h_φ(x,Z)$ that satisfies the constraint for every latent realization. Together these components define the predictive law itself, including its support and boundary probabilities, rather than serving as a final feasibility wrapper. We study how feasible coordinates and maps affect stochastic dimension, dependence, calibration, expressiveness, and computation. Experiments use Gaussian latent laws and fixed geometry-derived maps across affine equalities, ordering and simplex constraints, nonlinear manifolds, and three structural representations of seven-basin hydrological flow-duration-curve (FDC) data. In an official-source affine comparison with ProbHardE2E/DPPL, both methods achieve zero practical constraint violations. SLP-ProbHard uses 8 instead of 11 stochastic coordinates and improves MSE/MAE, while DPPL yields better marginal CRPS and closer-to-nominal coverage; a paired test detects no Energy Score difference across ten seeds. Real-world affine and nonlinear FDC representations reduce 13 to 7 and 14 to 8 ambient versus computational coordinates, respectively. Exact feasibility alone thus does not determine a predictive law, motivating direct structural generation when meaningful feasible coordinates are available.

Figures & tables

Tested geometry	Synthetic benchmark	Real-world representation	Primary SFLP	Hard comparator
Linear equality	hierarchy $Ay=b$	affine FDC coherence	$y_{0}+Nz$ ; affine $(Q,\operatorname{diff}Q)$	orthogonal projection; DPPL official-source adapter (synthetic only)
Convex structural instances	weak order; simplex	nonnegative FDC ordering	cumulative / normalized positive-part	isotonic / simplex projection
Explicit nonlinear equality/manifold	unit circle	FDC scale-shape $\Delta_{i}=RP_{i}$	angle / composed scale-shape	radial projection / alternating feasible correction

Table 1: Empirical constraint coverage. These experiments evaluate SLP-ProbHard on representative structures from the listed geometry groups. The direct official-source DPPL comparison is restricted to the synthetic affine benchmark.

Family	Method	MSE $\downarrow$	CRPS $\downarrow$	ES $\downarrow$	Cov.90	Stoch. dim.	Feasibility
Linear	Projection-Or	0.2965	0.2847	1.2088	0.785	11	VR ${}_{10^{-5}}=0$
Linear	DPPL (official-source adapter)	0.2997	0.2868	1.2190	0.809	11	VR ${}_{10^{-5}}=0$
Linear	SLP-ProbHard	0.2957	0.2901	1.2216	0.786	8	VR ${}_{10^{-5}}=0$
Order	Projection-isotonic	5.5677	0.7945	2.2440	0.784	5	exact order
Order	SLP-positive-part	5.0948	0.7820	2.1883	0.797	5	exact order
Simplex	Projection-simplex	0.0322	0.0890	0.2474	0.851	5	CE $1.51\!\times\!10^{-8}$

Table 2: Primary synthetic benchmarks (means across untouched seeds). “DPPL (official-source adapter)” denotes ProbHardE2E/DPPL using the official released source with the verified affine adapter; full source audit and paired statistics are in Appendix O .

Real-world representation	Hard method	Raw- $Q$ CRPS	Raw- $Q$ ES	Cov.90	Width.90	Stoch. dim.	Hard VR
Nonneg. order	Projection-nonneg-isotonic	3.894	15.561	0.757	14.15	7	0
	SLP-nonneg-positive-part	3.710	14.603	0.722	22.26	7	0
Affine coherence	End-to-end affine projection	3.627	13.649	0.746	13.57	13	0
	SLP-affine	3.698	13.997	0.712	10.94	7	0
Nonlinear scale-shape	Post-hoc feasible correction	3.568	13.754	0.823	14.97	14	0
	End-to-end feasible correction	4.177	14.483	0.630	15.75	14	0

Table 3: Three structural views of the same frozen seven-basin seasonal FDC dataset. “Hard VR” is the representation-specific

10^{-5}

violation rate; the nonlinear hard methods also have zero negative reconstructed discharge and zero ordering violations. For the affine rows, Hard VR measures increment-coherence equalities only.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Work	Core mechanism	Prob. law?	Hard feasibility	Representation emphasis
OptNet / diff. convex ( Amos and Kolter, 2017 ; Agrawal et al., 2019 )	optimization layer	opt.-centered	supported problem	optimization geometry
DC3 ( Donti et al., 2021 )	completion + correction	det./opt.	correction-dependent	completion/correction
Frerix et al. (2020)	direct linear-inequality map	deterministic	yes, supported class	feasible geometry
RAYEN ( Tordesillas et al., 2023 )	direct convex feasible layer	deterministic	yes, convex class	convex parameterization
CLOVER ( Olivares et al., 2024 )	differentiable coherent structural generation	yes	yes, hierarchy	coherent probabilistic representation
Girolimetto-Di Fonzo ( Girolimetto and Di Fonzo, 2024 )	free/constrained structural-like representation	yes	yes, linear coherence	general linear structural coordinates

Appendix

Table 4: Literature comparison by mechanism and probabilistic emphasis. Structural probabilistic generation has important precedents in hierarchical, linear, and domain-specific settings. SLP-ProbHard uses structural representation as a cross-family probabilistic model-design object and jointly analyzes its consequences for support/boundary probability, stochastic coordinates, expressiveness/calibration, and computation. The direct ProbHardE2E comparison in this paper is restricted to the compatible synthetic affine benchmark.

Constraint	Feasible set	Structural map	Qualification
Non-negativity	$\mathbb{R}_{+}^{d}$	$h_{i}(z)=\psi_{+}(z_{i})$	$\psi_{\mathrm{bm}}$ : boundary mass; $\psi_{\mathrm{br}}$ : reachable boundary; $\psi_{\mathrm{int}}$ : interior only.
Bounded interval	$[a,b]^{d}$	$a+(b-a)\sin^{2}(z_{i})$ or $a+(b-a)\sigma(z_{i})$	$\sin^{2}$ reaches endpoints; sigmoid excludes them.
Weak ordering	$y_{1}\leq\cdots\leq y_{d}$	$y_{1}=z_{1}$ , $y_{k}=y_{k-1}+\psi_{+}(z_{k})$	Nonnegative increments; boundary behavior depends on $\psi_{+}$ .
Ordering implementations	same	$\psi_{\mathrm{bm}}(z)=\left[z\right]_{+}$ or $\left[z\right]_{+}^{p}$ ; $\psi_{\mathrm{br}}(z)=z^{2}$ ; $\psi_{\mathrm{int}}(z)=\softplus(z)$	Positive-part family gives tie mass; square is boundary-reachable; softplus is interior.
Linear equality	$Ay=b$	$h(z)=y_{0}+Nz$ , $AN=0$ , $Ay_{0}=b$	Exact and surjective for a full null-space basis; latent dimension $d-m$ .
Simplex	$\Delta^{d-1}$	$a_{i}=\psi_{+}(z_{i})$ , $y_{i}=a_{i}/\sum_{j}a_{j}$ when active; feasible fallback if all inactive	Normalized nonnegative structural map; face mass depends on $\psi_{+}$ .

Appendix

Table 5: Representative Structural Feasibility Maps. Coverage, boundary reachability, and boundary probability are distinct properties.

Structural map	Predictive-law structure	Moment propagation	Boundary characterization
Affine equality	affine pushforward	exact mean/covariance	feasible affine manifold
Positive-part boundary-mass	atom at zero + continuous positive component	exact for Gaussian latent	$\Pr(Y=0)=\Phi(-\mu/\sigma)$
Weak ordering	cumulative mixed law	exact for independent Gaussian coordinates	exact adjacent-tie probabilities
Nonnegative ordering	cumulative mixed law	exact for independent Gaussian coordinates	exact zero and tie probabilities
Square	squared-Gaussian transform	exact Gaussian moments	boundary reachable, zero mass
Boundary-mass normalized simplex	mixture over simplex faces	conditional/numerical moments	analytic active-set probabilities

Appendix

Table 6: Distributional propagation under representative structural maps. Analytic tractability is separate from hard feasibility: every listed map is defined through its pushforward law, while closed-form summaries are used when available.

Method	Primary mechanism	Probabilistic predictive law	Boundary behavior	Main qualification
DC3 ( Donti et al., 2021 )	Equality completion plus iterative inequality correction	Not a dedicated probabilistic framework	Completion or correction dependent	Requires correction machinery.
Frerix et al. (2020)	Direct parameterization for homogeneous linear inequalities	Deterministic focus	Represented feasible boundary available	Specialized constraint class.
ProbHardE2E ( Utkarsh et al., 2026 )	Differentiable probabilistic projection	Yes	Determined by projected law	Broad supported constraint classes; projection machinery required.
Soft-Radial ( Schneider and Kuhn, 2026 )	Radial map into convex-set interior	Not central	Interior-oriented	Requires convex geometry and an interior anchor.
SLP-ProbHard	Stochastic latent law plus structural feasibility map	Yes	Map-dependent: excluded, reachable-only, or positive mass	Requires a suitable structural map.

Appendix

Table 7: Representative mechanisms. Several approaches provide hard feasibility; SLP-ProbHard is distinguished by making a constraint-specific structural map part of a probabilistic pushforward model.

Criterion	Projection / ProbHardE2E family	SLP-ProbHard	Implication
Core mechanism	Unconstrained probabilistic representation followed by projection/correction	$Y=h(Z)$ with $h(\mathcal{Z})\subseteq\mathcal{C}$	Different constructions of the feasible law.
Organizing taxonomy	Linear/nonlinear equalities and convex inequalities under projection	Affine, nonnegative/ordered, normalized/compositional, composed, and functional structural families	Different axes of generality.
Constraint generality	Broad for supported equality and convex-inequality classes	Selective and map-dependent	Projection is generally broader.
Linear equality coordinates	Ambient prediction before projection	$q=d-m$ free coordinates	Exact intrinsic reduction in this family.
Nonlinear coordinates	Ambient or method-specific	May be intrinsic or redundant	Dimensional benefit is not universal.
Projection metric	Selected geometry may matter	No projection metric for explicit maps	Removes metric choice.

Appendix

Table 8: Mechanistic comparison with probabilistic projection.

Frozen item	Value used in the primary synthetic benchmarks
Train / validation / test size	3000 / 750 / 750
Backbone	MLP, hidden width 128, depth 3, dropout 0
Batch size	128
Maximum epochs / early stopping	60 / patience 20, minimum improvement $10^{-6}$
Learning rate / weight decay	$10^{-3}$ / $10^{-5}$
Gradient clipping	5.0 where exposed by the frozen runner

Appendix

Table 9: Primary synthetic training configuration. Competing methods share the same backbone/training protocol within each benchmark; only the constraint mechanism and corresponding output/latent head differ where required.

Benchmark	Frozen target-generation structure	Same structure as primary SLP-ProbHard map?	Potential SLP-ProbHard alignment	Intended scientific question
Affine	Feasible latent regression is mapped through a fixed affine null-space representation $y=y_{0}+Nz$ .	Yes	Yes: coordinates match the feasible DGP	Can a probabilistic model learn only the $q=d-m$ free directions while matching ambient projection?
Weak ordering	A base response is combined with cumulative nonnegative positive-part increments, producing exact-tie mass.	Yes	Yes: boundary mechanism is aligned	Does a boundary-mass ordering representation change tie probability and proper scores relative to isotonic projection?
Simplex	Nonnegative components with an exact-zero mechanism are normalized to the simplex, producing face-supported observations.	Yes, at the structural level	Yes: boundary-capable normalization is aligned	How do direct boundary-capable normalization and Euclidean simplex projection differ in face probability and predictive scoring?
Circle	A conditional angle is drawn from a von Mises law and mapped as $(\cos\Theta,\sin\Theta)$ . Learned SLP-ProbHard uses a Gaussian angle; projection uses an ambient bivariate Gaussian.	Geometry yes; latent family no	Partial	Does one structural angle provide competitive hard-feasible prediction relative to ambient radial projection under a non-Gaussian angular DGP?
Composed nonneg. order	Targets are generated directly by cumulative positive-part from the first coordinate onward.	Yes, exactly	Yes, deliberately strong	Mechanism validation: do the predicted zero/tie boundary effects follow the preimage analysis when the structural DGP is known?

Appendix

Table 10: Synthetic-DGP transparency. The table separates structural alignment from the question each benchmark is designed to test. The composed benchmark is deliberately favorable and is interpreted only as mechanism validation.

Method	CEabs FP32	CEabs FP64	CEmax FP32	CEmax FP64	VR FP32	VR FP64
Projection-Or	$4.22\times 10^{-7}$	$6.89\times 10^{-16}$	$4.29\times 10^{-6}$	$4.44\times 10^{-15}$	0.0000	0.0000
Projection-Ob	$5.68\times 10^{-7}$	$1.28\times 10^{-15}$	$2.29\times 10^{-5}$	$1.89\times 10^{-13}$	0.0017	0.0000
SLP-ProbHard	$2.69\times 10^{-7}$	$4.97\times 10^{-16}$	$2.86\times 10^{-6}$	$5.33\times 10^{-15}$	0.0000	0.0000

Appendix

Table 11: Seed-20 diagnostic. Trained models are unchanged; only feasibility-map / residual computation is repeated in float64.

Method	MSE	MAE	CRPS	ES	$P(y_{1}=0)$	Pair-tie rate
Projection-nonneg-isotonic	$0.48397\pm 0.00507$	$0.49376\pm 0.00177$	$0.35222\pm 0.00122$	$0.93289\pm 0.00328$	$0.5803$	$0.5440$
SLP-nonnegative-positive-part	$0.48433\pm 0.00751$	$0.49437\pm 0.00413$	$0.35027\pm 0.00315$	$0.92047\pm 0.00740$	$0.5802$	$0.4374$
Target	–	–	–	–	$0.5850$	$0.4798$

Appendix

Table 12: Synthetic composed constraint benchmark, seeds 115-119. Both hard methods have zero negative values and zero ordering violations. SLP-ProbHard is closer on adjacent-tie frequency, whereas projection is closer on the anchor-zero frequency.

Method	MSE	MAE	CRPS	ES	Cov.90	Width.90	CEabs	VR ${}_{10^{-5}}$
Unconstrained	0.3622	0.4857	0.3422	0.5391	0.835	1.593	0.623	1.0000
Soft penalty	0.3613	0.4850	0.3438	0.5417	0.810	1.484	0.584	1.0000
Post-hoc radial	0.3649	0.4827	0.3344	0.5253	0.807	1.496	$4.80\times 10^{-8}$	0
End-to-end radial	0.3557	0.4893	0.3277	0.5147	0.864	1.704	$4.81\times 10^{-8}$	0
SLP-angle	0.3557	0.4836	0.3272	0.5143	0.854	1.611	$1.70\times 10^{-8}$	0

Appendix

Table 13: Unit-circle primary results, seeds 130-139. Residual differences between the two hard maps are numerical roundoff; both have zero

10^{-5}

violations. SLP-ProbHard uses one stochastic coordinate versus two for radial projection.

Method	Raw MSE	Raw MAE	Raw CRPS	Raw ES	Cov.90	Width.90	CEabs	VR ${}_{10^{-5}}$
Unconstrained	104.47	4.738	3.689	13.882	0.766	13.86	0.1270	1.000
Soft affine	104.37	4.731	3.686	13.882	0.763	13.62	0.1250	1.000
Post-hoc affine projection	103.72	4.685	3.717	14.099	0.681	9.90	$3.51\times 10^{-8}$	0
End-to-end affine projection	106.40	4.707	3.627	13.649	0.746	13.57	$5.85\times 10^{-8}$	0
SLP-affine	105.14	4.649	3.698	13.997	0.712	10.94	0	0

Appendix

Table 14: Real-world affine FDC coherence, seeds 150-159. Raw-

Q

metrics are primary; the affine branch enforces only

Q_{i}-Q_{i+1}+\Delta_{i}=0

, not nonnegativity/order.

Metric	Projection	SLP-ProbHard	Mean diff.	95% CI	paired $t$ $p$	Wilcoxon $p$
Raw MSE	106.398	105.136	-1.262	[-10.616, 8.092]	0.767	0.770
Raw MAE	4.707	4.649	-0.058	[-0.194, 0.078]	0.361	0.432
Raw CRPS	3.627	3.698	+0.072	[-0.032, 0.176]	0.153	0.193
Raw ES	13.649	13.997	+0.348	[-0.051, 0.747]	0.080	0.105
Coverage.90	0.746	0.712	-0.034	[-0.061, -0.0068]	0.0197	0.0195
Width.90	13.567	10.937	-2.630	[-3.075, -2.186]	$3.00\times 10^{-7}$	0.00195

Appendix

Table 15: Paired real-world affine comparison: SLP-ProbHard minus end-to-end affine projection. Paired tests did not detect differences in the principal accuracy/proper-score metrics at ten seeds; interval width and coverage show a significant sharpness-calibration trade-off.

Method	Raw MSE	Raw MAE	Raw CRPS	Raw ES	Cov.90	Width.90	Bilinear CE	Hard VR
Unconstrained	103.39	4.892	3.850	14.292	0.804	14.19	0.09695	1.000
Soft nonlinear	103.93	4.971	3.921	14.488	0.805	14.62	0.09597	1.000
Post-hoc feasible correction	102.33	4.840	3.568	13.754	0.823	14.97	0	0
End-to-end feasible correction	104.92	5.151	4.177	14.483	0.630	15.75	0	0
SLP-scale-shape	106.17	5.175	4.047	14.313	0.649	15.67	0	0

Appendix

Table 16: Real-world nonlinear scale-shape FDC coherence, seeds 170-179. All three hard methods also have zero negative reconstructed discharge and zero ordering violations.

Metric	Correction	SLP-ProbHard	Mean diff.	95% CI	paired $t$ $p$	Wilcoxon $p$
Raw MSE	104.916	106.166	+1.250	[-3.778, 6.279]	0.588	0.432
Raw MAE	5.151	5.175	+0.024	[-0.175, 0.223]	0.790	0.432
Raw CRPS	4.177	4.047	-0.130	[-0.285, 0.025]	0.090	0.232
Raw ES	14.483	14.313	-0.171	[-0.609, 0.268]	0.402	0.557
Coverage.90	0.630	0.649	+0.020	[-0.066, 0.105]	0.619	0.375
Width.90	15.747	15.669	-0.078	[-1.909, 1.753]	0.925	1.000

Appendix

Table 17: Paired real-world nonlinear comparison: SLP-ProbHard minus the five-iteration end-to-end alternating feasible correction. Paired tests did not detect differences in the principal predictive metrics at ten seeds.

Family	Metric	Reference mean	SLP-ProbHard mean	Rel. SLP-ProbHard diff.	95% CI of paired diff.	paired $t$ $p$	Wilcoxon $p$
Linear equality	mse	0.29646	0.29574	-0.25%	[-0.003834, 0.00238]	0.609	0.77
Linear equality	mae	0.39286	0.39295	+0.02%	[-0.001766, 0.001944]	0.916	0.77
Linear equality	crps	0.28467	0.29012	+1.91%	[0.003935, 0.00696]	1.91e-05	0.00195
Linear equality	energy score	1.20878	1.22158	+1.06%	[0.005968, 0.01964]	0.00218	0.00391
Weak ordering	mse	5.56771	5.09482	-8.49%	[-1.049, 0.1034]	0.0964	0.084
Weak ordering	mae	1.09838	1.08457	-1.26%	[-0.02347, -0.004143]	0.0103	0.0195

Appendix

Table 18: Paired untouched-seed comparisons. Relative difference is

100(\mathrm{SLP}-\mathrm{reference})/\mathrm{reference}

; negative values favor SLP-ProbHard for error/score metrics.

p

-values are unadjusted and are reported as diagnostics rather than as a multiple-comparison discovery exercise.

Method	MSE	CRPS	ES	Cov.90	Parameters	$\mathrm{VR}_{10^{-5}}$
Projection-Or	0.29741	0.28544	1.21124	0.7825	37,526	0
SLP-ProbHard diagonal	0.29543	0.28887	1.21714	0.7954	36,752	0
SLP-ProbHard low-rank	0.29591	0.28364	1.20710	0.8016	40,880	0
SLP-ProbHard full	0.29587	0.28376	1.20710	0.7969	40,364	0

Appendix

Table 19: Covariance expressiveness ablation, seeds 40-44. The experiment compares covariance families within SLP-ProbHard against the standard Projection-Or reference; it is not a matched full-covariance projection study.

Family	$d$	Reference ms	SLP-ProbHard ms	Ref./ SLP-ProbHard	Latency reduction
Linear equality	11	1.784	1.367	1.30 $\times$	23.4%
Ordering	5	248.802	0.958	259.7 $\times$	99.61%
Ordering	40	504.658	3.872	130.3 $\times$	99.23%
Simplex	5	1.471	1.382	1.06 $\times$	6.0%
Simplex	40	7.492	3.540	2.12 $\times$	52.8%
Circle	2	0.616	0.503	1.22 $\times$	18.3%

Appendix

Table 20: Controlled end-to-end CPU profiling: matched MLP + diagonal Gaussian sampling + feasibility operation, batch 112 and 100 predictive samples, 8 threads, float32, 5 warm-ups and 20 measured repeats. Ordering/PAVA ratios are implementation-level.

$d$	$m$	$q$	Dim.red.	$\Delta$ MSE	$\Delta$ CRPS	$\Delta$ ES	VR Proj.	VR SLP-ProbHard	CEabs Proj.	CEabs SLP-ProbHard
10	2	8	20.0%	+0.46%	+3.56%	+2.00%	0.0000	0.0000	$2.81\times 10^{-7}$	$3.55\times 10^{-7}$
11	3	8	27.3%	+0.71%	+2.06%	+1.22%	0.0000	0.0000	$4.13\times 10^{-7}$	$2.63\times 10^{-7}$
13	5	8	38.5%	-0.56%	+0.58%	+0.36%	0.0000	0.0000	$3.78\times 10^{-7}$	$2.19\times 10^{-7}$
19	3	16	15.8%	+0.03%	+2.45%	+1.11%	0.0000	0.0000	$5.53\times 10^{-7}$	$6.11\times 10^{-7}$
21	5	16	23.8%	-0.57%	+1.08%	+0.38%	0.0000	0.0000	$4.19\times 10^{-7}$	$4.00\times 10^{-7}$
25	9	16	36.0%	-0.19%	+0.46%	+0.36%	0.0000	0.0000	$4.78\times 10^{-7}$	$3.91\times 10^{-7}$

Appendix

Table 21: Twelve held-out linear scaling configurations, seeds 30-34.

$d$	MSE gain	CRPS gain	ES gain	Obs.tie	SLP-ProbHard tie	Proj.tie	SLP-ProbHard tie err.	Proj tie err.
3	+5.25%	+0.55%	+1.14%	0.261	0.181	0.300	0.079	0.040
5	+17.78%	+2.35%	+3.05%	0.251	0.235	0.401	0.017	0.149
10	+11.65%	+3.91%	+4.79%	0.255	0.234	0.504	0.021	0.249
20	+10.41%	+3.85%	+4.48%	0.254	0.232	0.591	0.022	0.337
40	+17.75%	+25.96%	+23.50%	0.253	0.241	0.891	0.012	0.638

Appendix

Table 22: Ordering scaling, seeds 70-74.

$d$	MSE gain	CRPS gain	ES gain	Obs.zero	SLP-ProbHard zero	Proj.zero	SLP-ProbHard zero err.	Proj zero err.
3	+0.85%	+2.80%	+2.50%	0.245	0.164	0.195	0.081	0.050
5	+1.51%	+2.80%	+1.87%	0.250	0.164	0.233	0.086	0.017
10	+0.82%	+3.38%	+1.54%	0.250	0.253	0.325	0.003	0.075
20	+2.53%	+4.47%	+2.16%	0.252	0.338	0.404	0.086	0.152
40	+1.99%	+4.76%	+1.82%	0.252	0.372	0.453	0.120	0.202

Appendix

Table 23: Simplex scaling, seeds 110-114. Positive gains mean lower SLP-ProbHard score.

Audit quantity	Result
Adapter vs. direct-reference mean, max abs. diff.	$0$
Adapter vs. direct-reference covariance, max abs. diff.	$0$
Conditioned-mean constraint max abs. residual	$1.19\times 10^{-7}$
$A\Sigma_{c}$ max abs. residual	$4.84\times 10^{-8}$
Float64 sample constraint max abs. residual	$2.66\times 10^{-15}$
Float32 sample CE abs	$4.60\times 10^{-8}$

Appendix

Table 24: Frozen equivalence and feasibility audit for the official-source affine adapter.

Metric	DPPL source adapter	SLP-ProbHard
MSE	$0.29968\pm 0.00797$	$\mathbf{0.29574\pm 0.00614}$
MAE	$0.39466\pm 0.00479$	$\mathbf{0.39295\pm 0.00360}$
Marginal CRPS	$\mathbf{0.28676\pm 0.00367}$	$0.29012\pm 0.00277$
Energy Score	$1.21896\pm 0.01550$	$1.22158\pm 0.01228$
Coverage 90	$\mathbf{0.8093\pm 0.0096}$	$0.7856\pm 0.0078$
Interval width 90	$1.2967\pm 0.0358$	$\mathbf{1.2637\pm 0.0232}$

Appendix

Table 25: Official-source ProbHardE2E/DPPL affine-adapter comparison versus the already-frozen SLP-ProbHard primary, seeds 20-29. Bold marks lower error/score, coverage closer to the nominal 0.90 level, smaller coordinate/parameter counts, and lower measured latency. Interval width is a sharpness diagnostic and must be interpreted jointly with coverage; latency is implementation-specific.

Metric	DPPL mean	SLP-ProbHard mean	SLP $-$ DPPL	95% CI	paired $t$ $p$	Wilcoxon $p$
MSE	0.29968	0.29574	$-0.00395$	$[-0.00653,-0.00137]$	0.0072	0.0098
MAE	0.39466	0.39295	$-0.00171$	$[-0.00328,-0.00015]$	0.0353	0.0645
CRPS	0.28676	0.29012	$+0.00336$	$[0.00198,0.00474]$	$3.76\times 10^{-4}$	0.0020
Energy Score	1.21896	1.22158	$+0.00263$	$[-0.00397,0.00922]$	0.3913	0.5566
Coverage 90	0.8093	0.7856	$-0.02367$	$[-0.03246,-0.01488]$	$1.81\times 10^{-4}$	0.0039
Width 90	1.2967	1.2637	$-0.03302$	$[-0.06293,-0.00310]$	0.0341	0.0371

Appendix

Table 26: Paired direct-comparator statistics across the same ten seeds.

p

-values are descriptive and unadjusted. Residual differences are numerical-scale effects: both methods have VR

{}_{10^{-5}}=0

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Aug 9, 2026cs.LG

Constrained Learning with Universally Learnable Concept Classes

We study constrained statistical learning over infinite-dimensional hypothesis classes in the fully nonconvex setting, and establish universal PACC learnability of the solutions of dual algorithms: Probably Approximately Correct on Constraints, guaranteeing optimality and constraint satisfaction at once. This strengthens near-PACC results, whose feasibility residual no amount of data can remove. Optimality is caught between generalization, governed by Rademacher complexity and favoring small classes, and strong Lagrangian duality, which rests on Lyapunov convexity for vector measures and needs decomposability, a demand pulling the other way. We reconcile the two by posing the population problem over a universal RKHS

\mathcal{H}_K

, dense in a decomposable envelope, and learning over norm balls of growing radius. This yields the Tikhonov complexity

\mathfrak{T}^{\varepsilon}_{n}

, the least RKHS norm reaching an

\varepsilon

-optimal Lagrangian level set; we prove it finite, obtain exact learnability of the optimal value, and make the sample threshold explicit and polynomial in

1/\varepsilon

under a source condition. Feasibility is harder: absent convexity the Lagrangian may not attain its infimum, and dual information pins down only an averaged constraint-risk vector, not the risks of any returned predictor. We introduce the closure-realization gap

\varepsilon^\star_\infty

, an index of how well

\mathcal{H}_K

retrieves feasible solutions from dualization; it is a property of the problem, not of a modeling choice. Learnability is exact when

\varepsilon^\star_\infty=0

, in particular under dual differentiability, and near-PACC with residual exactly

\varepsilon^\star_\infty

otherwise. Finally, no distribution-free threshold exists already in the unconstrained specialization, so universality is the canonical frame for dual algorithms over large hypothesis classes.

Herlock SeyedAbolfazl Rahimi, Spyridon Pougkakiotis, Dionysis Kalogerias

1Yale University · 2King’s College London

Learnability Uncoupled Learning Primal-Dual Methods Hypothesis Class Nonconvex Lyapunov Function

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Sep 26, 2026cs.AI

Prediction Limits and Koopman Closure of Geometry-Induced Soft State Abstractions

A soft state representation assigns each state a vector of nonnegative class weights that sum to one. We study how the construction of these weights and the state dynamics jointly determine the accuracy of linear prediction. For any fixed measurable representation, we derive a finite-sample lower confidence bound on the smallest population root-mean-square prediction error among matrices with a specified spectral-norm limit. The bound compares variation in successor coordinates within each reference class with the improvement that soft inputs could provide. It is computed from independent evaluation pairs without fitting a prediction matrix. A bound above a chosen tolerance rules out that tolerance for the entire matrix class; a zero bound is inconclusive. For coordinates constructed using Kernel Affine Hull Machines, reconstruction-score margins control disagreement with reference labels and enter bounds on prediction error. Under exact deterministic linear evolution, we also establish the Koopman and reproducing-kernel Hilbert-space adjoint interpretation, accounting for redundant coefficient vectors. A four-state study compares the confidence bound with analytically known optima across 117,000 reported replicate datasets. A Van der Pol representation selected on pilot data is then evaluated on 32 independent datasets under each of two transition laws. The reported bounds are positive at the fitted matrix norm, but can become zero at larger norm limits. Further forecasting studies examine coordinate variation, common prediction targets, and long-horizon error. The results distinguish agreement with reconstruction classes, attainable prediction accuracy, and exact operator closure.

Mohit Kumar, Somayeh Kargaran

Software Competence Center Hagenberg, Hagenberg, Upper Austria, Austria

Koopman Operator Next-State Prediction

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Jun 19, 2026cs.LG

Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modelling

Machine learning models are increasingly used to model chemical process systems, yet they often lack principled uncertainty quantification and mechanisms to enforce physical constraints. We propose a probabilistic neural network framework that guarantees satisfaction of linear equality constraints within a given tolerance, while capturing aleatoric uncertainty. Compared to state-of-the-art methods, our formulation demonstrates improved predictive accuracy, uncertainty calibration, and adherence to constraints on reduced data. It also demonstrates competitive performance, but with significantly faster training times when evaluated on large data regimes. We evaluated this on two batch reactor case studies, enforcing mass balances.

Matthew Marsh, Benoit Chachuat, Antonio del Rio Chanona

The Sargent Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, UK

Inequality Constraints Chemical Process Systems Parametric Physics-Informed Neural Network Calibrated Uncertainty Uncoupled Learning Neural Network

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