GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models

Khoa Nguyen; Daniel Serino; Aviral Prakash; Marc Klasky

cs.LGAug 21, 2026

GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models

Authors: Khoa Nguyen, Daniel Serino, Aviral Prakash, Marc Klasky

Organizations: Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Abstract

Neural-network surrogate models are increasingly used to accelerate scientific simulations, but their deployment in extrapolative and autoregressive settings requires input-dependent estimates of prediction error. In this work, we introduce GeoQ (Geometry-Aware Conditional Quantile Error Estimation), a non-intrusive calibration framework for estimating surrogate error at individual query points. GeoQ represents the error at a query point as an anchor-averaged calibration error plus a learned nonnegative correction. This correction is modeled as an upper conditional quantile of the anchor-relative error increment, using geometry-based features that encode representation-space displacement and local support density. A cross-fitting procedure generates approximately out-of-sample calibration tuples, while a feature-space k-nearest-neighbor support score identifies regions \textcolor{black}{where the learned error model is supported by calibration data}. We evaluate GeoQ on scalar regression, chaotic dynamics, medium-range weather forecasting, and Richtmyer-Meshkov instability prediction. The results demonstrate that geometry-aware conditional quantile modeling provides a practical and non-intrusive approach for validity-aware error estimation in scientific surrogate models.

Figures & tables

Method class	Applicability domain/	OOD-risk	Non-intrusive	Training	Inference
	awareness	detection		Scalability	Scalability
Bayesian neural network	No	Weak-Moderate	No	Medium	Medium
Deep ensemble	No	Moderate	Yes	Medium ∗	Medium ∗
Conformal prediction	No	Weak	Yes	High	High
Generative model	No	Weak	No	Medium	Medium
Gaussian process	Yes	Strong	No	Low	Low

Table 1: Qualitative comparison of common approaches for uncertainty and error estimation in scientific surrogate models. Applicability-domain awareness indicates whether the method explicitly incorporates distance or support information relative to training or calibration data. OOD-risk detection refers to the ability to flag queries outside the training or calibration support. Non-intrusive indicates whether the method can be applied to a pretrained surrogate without modifying its architecture or training objective. Entries summarize typical implementations; individual variants may differ.

Panel of Figure 1 — Figure 1: Schematic of the anchor-based GeoQ error model. A query point is compared with its local anchors in the representation space induced by $\phi$ . The anchor errors define a local baseline $\overline{E}_{k_{a}}(x)$ , and GeoQ learns a nonnegative correction $\widehat{R}_{k_{a}}(x)$ to estimate the query error.

Panel of Figure 2 — Figure 2: Schematic of the GeoQ cross-fitted calibration procedure.

Figure 3: Forrester equation: Visualization of the surrogate prediction error and the estimated error by the proposed GeoQ approach.

Panel of Figure 4 — Figure 4: KS equation: Visualization of the surrogate’s prediction errors and the estimated errors by the proposed GeoQ approach.

Panel of Figure 5 — Figure 5: WeatherBench dataset: Visualization at day $10$ of the surrogate prediction and the estimated error by the proposed GeoQ approach.

Panel of Figure 6 — Figure 6: RMI problem: Visualization of the surrogate prediction and the estimated error by the proposed GeoQ approach at different times for the median test case. The ground truth and the surrogate prediction are binary images for RMI evolution. The field represents the volume fraction of heavy gas, with $1$ denoting regions occupied by heavy gas and $0$ denoting regions with light gas.

Dataset	Method	Pinball $\downarrow$	Correlation $\uparrow$	Coverage $\uparrow$	Train. Time $\downarrow$	Inf. Time $\downarrow$
Forrester eq.	GP	0.001	0.799	1.000	20 s	0.02 s
	MC Dropout	0.300	0.977	0.150	40 s	4 s
	Deep Ens.	0.303	0.984	0.487	400 s	4 s
	Conf. Pred.	0.298	-	0.490	30 s	0.6 s
	GeoQ (ours)	0.117	0.989	0.804	500 s	1.2 s
KS eq.	GP	0.021	0.551	0.719	158 min	9 min

Table 2: Global performance comparison on the entire testing set, including both near-support and out-of-distribution samples. The proposed GeoQ framework is evaluated without restricting predictions to the learned safe region. Best results are shown in bold and second-best results are underlined.

Problem	Safe Fraction	Pinball safe	Correlation safe	Coverage safe
Forrester equation	0.582	0.000	0.978	1.000
KS equation	0.900	0.017	0.495	0.891
WeatherBench	1.000	0.006	0.763	0.902
RMI	0.959	0.001	0.502	1.000

Table 3: Validity-aware performance of the proposed method inside the learned safe region. Coverage safe and Pinball safe are computed only on samples satisfying

S(\psi)\leq\tau_{\mathrm{safe}}

. Higher Coverage and Safe Fraction are better, lower Pinball is better.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Choice	Pinball	Correlation	Coverage
$\phi=\operatorname{Id}_{\mathcal{X}}$	0.024	0.412	0.856
$\phi=\widehat{F}_{\mathrm{encoder}}$	0.019	0.490	0.868
$\psi=[\Delta z_{k_{a}},\Delta d_{k}]$	0.019	0.490	0.868
$\psi=[\|\Delta z_{k_{a}}\|,\Delta d_{k}]$	0.021	0.483	0.835
$\psi=[\|\|\Delta z_{k_{a}}\|\|_{2},\Delta d_{k}]$	0.020	0.481	0.868
$\psi=\Delta z_{k_{a}}$	0.021	0.478	0.832

Appendix

Table 4: Comparison of different choices in the proposed approach on KS equation.

Method	Forrester eq.	KS eq.	WeatherBench	RMI
GP	0.093	0.429	-	-
MC Dropout	0.879	0.522	0.176	0.082
Deep Ens.	0.884	0.446	0.227	0.082
Conf. Pred.	0.945	0.492	0.172	0.087
GeoQ	0.868	0.484	0.168	0.083

Appendix

Table 5: Comparison of the surrogate model accuracy (in terms of relative

L^{2}

error) in different approaches.

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