Uncertainty-Aware Interpolation (UAI)

• Uncertainty-aware interpolation (UAI) is a framework that modulates interpolation, local reconstruction, or upsampling using uncertainty measures like Gaussian variance or confidence functions.
• It integrates uncertainty directly during model training, enforcing local class-consistency and decision-boundary smoothing, and improving cross-modal tasks such as depth super-resolution.
• UAI spans diverse applications from surrogate modeling with weighted Leja nodes to spatiotemporal biomass estimation and uncertainty-aware PDE solvers, enhancing both accuracy and calibration.

In the cited literature, uncertainty-aware interpolation (UAI) can be understood as a family of methods in which interpolation, upsampling, surrogate approximation, or local reconstruction is explicitly modulated by uncertainty, confidence, or calibration. Rather than treating interpolation as a purely geometric or numerical operation, these methods use uncertainty to determine the region over which consistency should hold, the auxiliary information that may be transmitted, the node locations used for approximation, or the width and reliability of predictive intervals. Representative formulations include Certainty Volume Prediction in unsupervised domain adaptation, Symmetric Uncertainty-aware Feature Transmission in color-guided depth super-resolution, weighted Leja interpolation for arbitrary parameter distributions, diffusion-based 3D interpolation for explanation supervision, spatiotemporal Attentive Neural Processes for GEDI biomass, and confidence-aware latent interpolation in PDE solvers (Ringwald et al., 2021, Shi et al., 2023, Loukrezis et al., 2019, Zhao et al., 2024, Young et al., 4 Apr 2026, Feng et al., 30 Jul 2025).

1. Core abstractions and design patterns

Across these works, uncertainty is not represented by a single canonical object. It may appear as a scalar feature-space spread parameter, a transformation-induced discrepancy map, an ensemble-derived epistemic scale, a predictive Gaussian variance, or a confidence function that mixes deterministic and stochastic representations. A plausible common abstraction is that UAI separates the interpolation operator from the trust assigned to local evidence.

Setting	Uncertainty mechanism	Interpolation target
CVP for UDA	Multivariate Gaussian in feature space with scalar $\sigma$	Local feature neighborhoods
SUFT for DSR	Symmetric Uncertainty from flip-induced feature discrepancy	RGB-to-depth feature transmission
Weighted Leja interpolation	Density-weighted nested nodes	Surrogates under arbitrary PDFs
DUE for 3D explanation supervision	Diffusion resampling variance and uncertainty imputation	Missing 3D annotation slices
Spatiotemporal ANP for GEDI	Context-conditioned predictive mean and variance	Sparse space-time biomass fields
LVM-GP for PDEs	Confidence-weighted interpolation between neural feature and GP prior	Latent function representation

Two distinctions recur. First, several methods estimate uncertainty intrinsically during training rather than post hoc. CVP states that its uncertainty is intrinsic, participates in training through the reparameterization trick, and governs boundary regularization; DUE estimates interpolation uncertainty through repeated diffusion sampling and then predicts it with an auxiliary Neural Process-based VAE; LVM-GP learns a confidence function that adaptively mixes a deterministic feature and a Gaussian process prior (Ringwald et al., 2021, Zhao et al., 2024, Feng et al., 30 Jul 2025). Second, multiple papers separate aleatoric and epistemic effects. The GEDI ANP models aleatoric uncertainty as Gaussian predictive variance and epistemic uncertainty through a stochastic latent variable, while UACQR explicitly distinguishes variability of $Y \mid X$ from uncertainty in estimating that conditional distribution (Young et al., 4 Apr 2026, Rossellini et al., 2023).

2. Feature-space interpolation and decision-boundary smoothing

A direct formulation of UAI in representation space appears in "Certainty Volume Prediction" (Ringwald et al., 2021). For an input $x^{(i)}$, the feature extractor $f_\mathrm{CNN}$ produces a feature mean $\mu^{(i)} \in \mathbb{R}^{D}$, and uncertainty is modeled as a multivariate Gaussian,

$\mathcal{N}(\mu^{(i)}, \Sigma^{(i)}),$

with isotropic covariance $\Sigma^{(i)} = (\sigma^{(i)} \mathbb{1})^2$. The model samples

$$\varphi^{(i)}_m \sim \mathcal{N}(\mu^{(i)}, (\sigma^{(i)} \mathbb{1})^2),$$

so that training is performed not only on the feature mean but also on a local certainty volume around it (Ringwald et al., 2021).

The mechanism is explicitly tied to interpolation-like local consistency. The sample consistency loss

$$\mathcal{L}_{ce}^{\varphi}(\hat{y}^{(i)}, \varphi^{(i)}) = -\frac{1}{M} \sum_{m=1}^M \sum_j \hat{y}^{(i)}_j \log(f_\mathrm{Cl}(\varphi^{(i)}_m)_j)$$

forces nearby samples to be predicted as the same class. An antagonistic loss,

$$\psi^{(i)} = \max(0,\, \kappa - \mathcal{L}_{ce}^{\varphi}(\hat{y}^{(i)}, \varphi^{(i)})),$$

combined with

$Y \mid X$0

prevents overconfident overspread in uncertain regions. The paper’s intuition is that features far from the decision boundary can have large $Y \mid X$1, while those close to boundaries require smaller $Y \mid X$2.

Theoretical and empirical interpretation are aligned. CVP directly operationalizes the cluster assumption and is linked to decision-boundary smoothness. Smoothness is evaluated with an oscillation metric in which features are linearly interpolated in feature space; frequent changes in predicted label indicate a rough boundary. CVP significantly reduces such oscillations, with approximately 14k fewer classification changes in boundary interpolations. The uncertainty proxy $Y \mid X$3 also shows strong Pearson correlations with max logit, entropy, and MC Dropout mean and variance, which the paper uses to argue that $Y \mid X$4 is a meaningful uncertainty measure. On benchmarks, CVP reports 90.1% on VisDA-2017 with a ResNet-101 backbone, 76% mean accuracy on Adaptiope, and consistent 2–3% average gains versus ablations without uncertainty modeling. Qualitative analysis associates low $Y \mid X$5 with out-of-distribution, occluded, broken, or unusual samples.

A common misconception is that uncertainty-aware interpolation in classification must be applied only at the output layer. CVP contradicts that view by locating uncertainty in feature space and using it to regularize the admissible local region around a representation. This suggests that, in discriminative settings, UAI can be interpreted as local class-consistency enforcement over uncertainty-shaped neighborhoods rather than as confidence calibration alone.

3. Cross-modal upsampling and 3D imputation

In color-guided depth super-resolution, UAI is instantiated as uncertainty-gated feature transmission. "Symmetric Uncertainty-aware Feature Transmission for Depth Super-Resolution" identifies two difficulties: the resolution gap between LR depth and HR RGB and the cross-modality gap by which RGB textures may be harmful for HR depth reconstruction (Shi et al., 2023). Standard interpolation such as bicubic is described as amplifying noise and blurring details. SUFT replaces common interpolated pre-upsampling with an iterative up-and-down sampling pipeline. At each fusion stage, the current depth features are upsampled by learned up-projection, a horizontally flipped copy is upsampled and realigned, and uncertainty is derived from the discrepancy

$Y \mid X$6

Channelwise average and max pooling are concatenated and normalized,

$Y \mid X$7

and the resulting uncertainty map gates RGB features before fusion:

$Y \mid X$8

The stated intuition is that ambiguous areas such as edges and textures receive stronger RGB guidance, whereas smooth or flat regions are less influenced, reducing erroneous texture transfer. SUFT reports state-of-the-art RMSE on NYU v2, Middlebury, and RGB-D-D, and its ablations show that iterative upsampling and Symmetric Uncertainty both improve RMSE, with the full model performing best (Shi et al., 2023).

A related but distinct formulation appears in "DUE: Dynamic Uncertainty-Aware Explanation Supervision via 3D Imputation" (Zhao et al., 2024). The setting is 3D medical imaging, where only sparse 2D annotation slices are available and uncertainty varies across different parts of the explanation. DUE combines 3D explanation supervision, distance-sensitive 3D interpolation, and uncertainty-aware explanation guidance. Missing slices between annotated planes are generated by a conditional diffusion model. The paper then re-samples the diffusion process multiple times and computes voxel-wise variance across sampled interpolations as a proxy for uncertainty. Because repeated sampling is expensive, DUE trains a Neural Process-based VAE as a fast uncertainty imputer and converts predicted uncertainty into weights that downweight ambiguous regions and upweight confident regions during explanation supervision. The final objective uses the interpolated mask multiplied by the uncertainty weight map.

The empirical claims are operational rather than purely formal. On pancreatic tumor classification and lung nodule classification, DUE reports the highest ROC-AUC and PR-AUC, and the best IoU, recall, and F1 for explanation maps. The paper states that DUE improves ROC-AUC by up to approximately 23% over 2D state-of-the-art methods on pancreatic tumor classification, and improves overlap with expert annotations by up to approximately 21% on pancreas and approximately 27% F1 on lung nodule classification. In both SUFT and DUE, the interpolated object is not merely a resized signal; it is a confidence-weighted reconstruction whose influence is spatially selective.

4. Arbitrary-distribution surrogate construction

In numerical uncertainty quantification, UAI appears as interpolation under nonstandard input distributions. "Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions using Weighted Leja Interpolation" argues that interpolation methods based on Lagrange interpolation are typically abandoned when one or more parameters do not follow normal, uniform, or closely related distributions, because interpolation nodes are difficult to define for general distributions (Loukrezis et al., 2019). The paper introduces weighted Leja nodes, defined by

$Y \mid X$9

where $x^{(i)}$0 is the one-dimensional input PDF and $x^{(i)}$1 is the support. The square root is used to ensure asymptotic correspondence with weighted Gaussian quadrature nodes.

The method’s principal numerical property is nestedness. Nodes are added one by one, so hierarchical Lagrange interpolation can be written incrementally through hierarchical surpluses. In the multivariate case, the interpolant is assembled on a downward-closed multi-index set $x^{(i)}$2 by a generalized sparse-grid combination,

$x^{(i)}$3

A dimension-adaptive greedy algorithm refines the index with the largest surplus contribution until an error tolerance or budget is reached. Because the node construction only requires PDF evaluations, the approach applies to truncated normal, Gumbel, and other arbitrary continuous distributions.

The uncertainty-aware character of this interpolation framework lies in how the surrogate is aligned to the probability law of the uncertain inputs. The resulting interpolant is used for expected value, variance, and higher moments, with weighted interpolatory means computed through products of one-dimensional weights. In four numerical experiments—the Borehole Model, Steel Column Reliability, Meromorphic Function, and Dispersive Waveguide—the paper compares weighted Leja interpolation with generalized Polynomial Chaos built by least-angle regression. Reported outcomes are either comparable or superior. The Borehole and Dispersive Waveguide examples show about an order-of-magnitude advantage in RMS error for Leja throughout much of the cost range, while the Steel Column Reliability example reports similar RMS behavior. In the Borehole example, for 500 function calls, Leja gives RMS error of approximately $x^{(i)}$4 and percent error in mean of approximately $x^{(i)}$5, versus approximately $x^{(i)}$6 and approximately $x^{(i)}$7 for gPC-LAR.

This line of work shows that UAI in computational science need not mean local feature smoothing. It can instead mean probability-aware placement of interpolation nodes so that surrogate accuracy and uncertainty statistics estimation remain reliable under arbitrary parameter distributions.

5. Spatiotemporal and physics-informed probabilistic interpolation

A spatiotemporal variant of UAI is developed in "Spatiotemporal Interpolation of GEDI Biomass with Calibrated Uncertainty" (Young et al., 4 Apr 2026). The problem is to infer spatially explicit and temporally continuous aboveground biomass density under irregular orbital sampling and long observational gaps, including a 13-month hibernation from March 2023 to April 2024. The paper extends the Attentive Neural Process to jointly sparse spatiotemporal settings and treats space and time symmetrically through the coordinate vector

$x^{(i)}$8

Inputs include this joint coordinate, a geospatial foundation model embedding from Tessera, and observed log(AGBD) for context points. The architecture comprises an embedding encoder, a context encoder, a deterministic path with multihead cross-attention, a stochastic path that models function-level epistemic uncertainty, and a decoder that outputs predictive mean and variance.

Training is episodic: within each tile, observed points in space and time are partitioned into context and target sets. The model is optimized with a conditional variational objective,

$x^{(i)}$9

Calibration is then assessed via standardized residuals, empirical 1$f_\mathrm{CNN}$0 and 2$f_\mathrm{CNN}$1 coverage, and stratification by disturbance intensity. A central empirical claim is that standard ensemble methods are systematically miscalibrated for GEDI biomass, with observed 68% coverage intervals capturing as little as 19% of actual errors, whereas the ANP produces Z-score standard deviation close to 1.0 across all regions and maintains credible intervals in disturbed tiles. The paper also reports that XGBoost can reach a Z-score standard deviation of 13 in Guaviare’s disturbed tiles. The proposed ANP is presented as producing well-calibrated uncertainty estimates across disturbance regimes.

A distinct latent-space formulation appears in "LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process" (Feng et al., 30 Jul 2025). Here the interpolation occurs between a learnable deterministic feature and a Gaussian process prior:

$f_\mathrm{CNN}$2

with $f_\mathrm{CNN}$3. The confidence function $f_\mathrm{CNN}$4 is learned from data. When $f_\mathrm{CNN}$5, the model relies on the deterministic feature; when $f_\mathrm{CNN}$6, it relies on the GP prior. The variance of the latent representation is proportional to $f_\mathrm{CNN}$7, so low-confidence regions inherit larger uncertainty. A neural-operator decoder then defines a conditional Gaussian distribution over the solution field, while physical laws are enforced as soft constraints through the loss.

Compared with Bayesian physics-informed neural networks and deep ensembles, LVM-GP is reported to achieve competitive predictive accuracy and robust uncertainty quantification, with mean predictions and standard deviations close to B-PINN-HMC in forward and inverse problems and more reliable behavior than deep ensembles as observation noise increases. A plausible implication is that, in PDE settings, UAI can be realized as confidence-aware interpolation in latent function space rather than only in observation space.

6. Calibration, visualization, and adjacent uncertainty-aware evaluation

A geometric and visualization-oriented form of UAI is presented in "Estimation and Visualization of Isosurface Uncertainty from Linear and High-Order Interpolation Methods" (Ouermi et al., 2024). The paper focuses on model uncertainty arising from interpolation in Marching Cubes. Linear interpolation is described as misplacing vertices, failing to recover sharp or subgrid features, and producing holes, broken pieces, and hidden features. To expose these errors, the paper derives an efficient analytical approximation of edge-crossing error,

$f_\mathrm{CNN}$8

based on Taylor expansion and divided differences. High-order cubic and WENO interpolation are then used both for more accurate crossings and for hidden feature detection and reconstruction. The associated visualization system includes error colormaps, binary thresholding, comparison surfaces, a CDF plot for threshold selection, local selection boxes, and transparent overlays for recovered features. Synthetic and real-world datasets are used to show that the analytical estimate closely matches true geometric error and identifies regions where high-order interpolation materially changes the isosurface.

Calibration-oriented extensions of uncertainty-aware prediction provide a complementary perspective. "Integrating Uncertainty Awareness into Conformalized Quantile Regression" separates aleatoric and epistemic uncertainty in prediction intervals (Rossellini et al., 2023). UACQR-S inflates the lower and upper quantiles using local epistemic scales $f_\mathrm{CNN}$9 and $\mu^{(i)} \in \mathbb{R}^{D}$0 estimated from ensembles, whereas UACQR-P uses ensemble order statistics directly. Both preserve the standard split-conformal marginal coverage guarantee,

$\mu^{(i)} \in \mathbb{R}^{D}$1

while empirically improving conditional coverage in regions of high epistemic uncertainty. This corrects a common simplification in which interval width is treated as an aleatoric proxy only.

"Interval-Based AUC (iAUC): Extending ROC Analysis to Uncertainty-Aware Classification" generalizes discrimination analysis to interval-valued predictions (Li et al., 4 Feb 2026). It defines

$\mu^{(i)} \in \mathbb{R}^{D}$2

which partition pairwise rankings into correct, incorrect, and ambiguous orderings. The framework supports selective prediction by abstaining on overlapping intervals and defines

$\mu^{(i)} \in \mathbb{R}^{D}$3

Under valid class-conditional coverage, $\mu^{(i)} \in \mathbb{R}^{D}$4 and $\mu^{(i)} \in \mathbb{R}^{D}$5 bound the optimal AUC up to the coverage-error term $\mu^{(i)} \in \mathbb{R}^{D}$6.

A more general inferential analogue appears in "Uncertainty-Aware Bayes' Rule and Its Applications" (Wang, 2023). The paper replaces standard Bayes’ rule with

$\mu^{(i)} \in \mathbb{R}^{D}$7

where the exponents rebalance trust in prior and likelihood depending on whether each is overly conservative or overly opportunistic. Applications include an uncertainty-aware Bayes classifier, Kalman filter, particle filter, and interactive-multiple-model filter. This suggests that UAI belongs to a broader methodological program in which interpolation, ranking, and posterior updating are all modified so that the operator itself responds to uncertainty rather than assuming fixed trust in its inputs.

Three clarifications follow from this literature. First, uncertainty awareness is not equivalent to post hoc calibration; several methods build uncertainty into the interpolation mechanism itself. Second, linear interpolation is not neutral: in isosurface extraction and depth super-resolution it can amplify noise, blur detail, or miss topology. Third, uncertainty should not be conflated with interval width alone; multiple works distinguish aleatoric variability from epistemic or contextual uncertainty and show that this distinction changes both training and evaluation.