Technical Summary: Locally Adaptive Conformal Inference for Operator Models

Problem Statement

Operator models are regression algorithms mapping between Banach spaces of functions ($\Gamma: \mathcal{F} \to \mathcal{G}$), widely used in spatiotemporal forecasting, physics emulation, and PDE approximation. A critical challenge in these applications is Uncertainty Quantification (UQ). Unlike scalar regression, operator models require function-valued prediction sets rather than scalar intervals.

Standard Conformal Prediction (CP) relies on the assumption of global exchangeability of data. However, in real-world functional data (e.g., climate, energy demand, weather), this assumption is frequently violated due to non-stationarity, seasonal drift, and varying input structures. While existing adaptive CP methods address non-stationarity by weighting calibration scores or modifying quantiles, they typically assume a global score geometry (e.g., residual norms). This approach fails for functional residuals where uncertainty is anisotropic and structured; norm-based scores produce spherical prediction regions that do not align with the true shape of residual variability.

Methodology: Local Sliced Conformal Inference (LSCI)

The authors propose Local Sliced Conformal Inference (LSCI), a distribution-free framework that generates locally adaptive, function-valued prediction sets. Unlike prior methods that localize the quantile, LSCI localizes the conformity score distribution itself.

Core Components

1. Local Exchangeability Framework:
   The method assumes the residual process is locally exchangeable (Campbell et al., 2019). This implies that while global distributions may drift, residuals associated with similar inputs (in a pre-metric space) share approximately exchangeable distributions.

2. $\Phi$-Depth Scores:
   Instead of using scalar norms, LSCI employs functional data depth to measure the centrality of a residual. Specifically, it uses $\Phi$-depths (Mosler & Polyakova, 2012), which are projection-based depths.

   • A residual $r$ is projected onto a family of linear functionals $\Phi$ (e.g., random Gaussian slices).
   • The depth is defined as the infimum of univariate depths (e.g., Tukey half-space depth) across these projections:
     $$D_\Phi(r | P) = \inf_{\phi \in \Phi} D(\phi(r) | \phi(P))$$
   • This allows the score to adapt to the anisotropic shape of the local residual cloud, rather than forcing an isotropic (spherical) geometry.

3. Local Score Construction:
   For a test input $f_{n+1}$, LSCI constructs a test-specific empirical residual distribution using similarity weights.

   • Knockoff Localization: To maintain finite-sample guarantees, weights are computed based on the distance between calibration inputs and a statistical knockoff of the test input ($\tilde{f}_{n+1} = f_{n+1} + \epsilon$), rather than the test input itself.
   • Feature Maps: Localization can occur in raw input space or a learned/transformed feature space (e.g., FPCA, Fourier modes) to capture semantic similarity.
   • Slice Variance Normalization: To handle heteroskedasticity, projected slices are rescaled using test-specific weights before depth calculation.

4. Prediction Set Formation:

   • The method computes local conformity scores for calibration residuals.
   • A threshold is determined by the $k$-th order statistic of these local scores (where $k = \lfloor \alpha(n+1) \rfloor$).
   • The prediction set is defined as the set of functions whose residuals lie within the corresponding central region of the local depth distribution:
     $$C_\alpha(f_{n+1}) = \{ \hat{\Gamma}(f_{n+1}) + r : D_\Phi(r | \hat{P}_{n+1}) \geq q_\alpha(f_{n+1}) \}$$

5. Sampling:
   Since the prediction set is an implicit subset of function space, the authors employ a sampling algorithm (Algorithm 1) using a locally adapted Functional Principal Component Analysis (FPCA) basis. Samples are drawn via inverse transform sampling on weighted projected coordinates and rejected if they fall outside the depth threshold.

Theoretical Guarantees

The paper establishes finite-sample validity under local exchangeability.

• Coverage Bound: The authors derive a data-dependent upper bound on the coverage gap. The gap is controlled by the total variation distance between the joint distribution of residuals and its permutations, which is bounded by the pre-metric distance between inputs (Proposition 3.1).
• Validity: Under full exchangeability, the method achieves exact marginal coverage. Under local exchangeability, the coverage remains close to the nominal level $1-\alpha$ provided the residual process varies smoothly with respect to the input pre-metric.

Experimental Results

The authors evaluated LSCI on synthetic Gaussian Process tasks and three real-world applications: Air Quality monitoring (PM2.5), Energy Demand forecasting, and Weather prediction (ERA5 temperature).

• Synthetic Tasks: LSCI achieved near-nominal coverage across various localization kernels, feature maps, and bandwidths. It demonstrated robustness to the choice of hyperparameters.
• Comparison with Baselines: Compared to standard conformal baselines (Conf), conformalized integrated bands (Supr), and deep operator models (PONet, QONet, UQNO), LSCI consistently produced tighter prediction sets (lower Interval Scores and Band Widths) while maintaining valid coverage.
  • In heteroskedastic settings, LSCI adapted to local variance structures, whereas global methods either over-covered (conservative) or under-covered.
• Real-World Applications:
  • Weather-ERA5: LSCI captured seasonal and regional error structures that global thresholds averaged out, resulting in significantly lower interval scores compared to non-adaptive methods.
  • Robustness: LSCI showed resilience against biased predictions and specific out-of-distribution noise regimes where baseline methods failed or required excessive interval expansion.
• Spatial Adaptivity: Visualizations confirmed that LSCI bands deform to match local residual anisotropy and seasonality, providing function-valued uncertainty that aligns with physical realities.

Significance and Claims

The paper claims that LSCI represents a significant advancement for operator learning by:

1. Bridging the Gap: It successfully applies conformal inference to functional data where standard norm-based scores fail due to anisotropy.
2. Localizing at the Score Level: By localizing the score distribution rather than just the quantile, LSCI models the geometry of uncertainty more accurately than existing adaptive CP methods.
3. Distribution-Free Guarantees: It retains the rigorous, distribution-free finite-sample guarantees of conformal inference, even in non-stationary environments, provided local exchangeability holds.
4. Practical Utility: The method provides practitioners with function-valued prediction sets that allow for the assessment of global reliability and the identification of regions with concentrated uncertainty, which is critical for high-stakes scientific and engineering applications.

Limitations

The authors acknowledge several limitations:

• Computational Overhead: Test-time localization and sampling (especially for high-resolution fields) introduce computational costs, though batching and GPU evaluation can mitigate this.
• Implicit Sets: The prediction set is defined implicitly via depth; practical application relies on Monte Carlo sampling. While stable in experiments, complex or high-resolution fields may require improved sampling mechanisms to avoid artifacts (e.g., overly concentrated samples).
• Smoothness Assumption: The method assumes smooth changes in the conditional distribution. Abrupt changes or changepoints may be oversmoothed by local kernels, leading to poor adaptivity in those specific regions.