Technical Summary: Perturbation-Based Conformal Prediction for Operator Learning in Data-Scarce Regimes

1. Problem Formulation

The paper addresses the challenge of Uncertainty Quantification (UQ) for neural operators learning the 2D incompressible Navier–Stokes equations. While neural operators (e.g., Fourier Neural Operators, FNO) provide efficient surrogates for expensive PDE solvers, their point predictions lack calibrated uncertainty estimates for spatiotemporal fields. This is critical in safety-critical or decision-making contexts where errors vary across space and time.

The specific problem is formulated as learning a solution operator $G: \mathbf{a} \to \mathbf{u}$, mapping an input history of vorticity fields $\mathbf{a}$ to a future trajectory $\mathbf{u}$. The goal is to construct a prediction set $C(\mathbf{a})$ that satisfies simultaneous coverage over the entire spatiotemporal grid with a probability of at least $1-\alpha$:
$$P(\mathbf{u}_{new} \in C(\mathbf{a}_{new})) \geq 1 - \alpha$$
The authors focus on the data-scarce regime, where the total label budget is fixed. In this setting, methods requiring separate uncertainty networks (like UQNO) must partition the limited data between a predictive model and an auxiliary uncertainty model, potentially reducing the performance of both.

2. Methodology: Perturbation-Based Nonconformity Score

The proposed framework wraps a trained neural operator with Split Conformal Prediction (CP). The core innovation lies in the construction of a perturbation-based nonconformity score that generates an adaptive, spatially varying uncertainty scale without requiring a separate uncertainty network or disjoint data splits.

2.1 Core Mechanism

The method relies on the principle that the sensitivity of a learned operator to small perturbations in training labels correlates with predictive uncertainty.

1. Dual Training: Two neural operators with identical architectures are trained on the same proper training set:
   • $\hat{G}$: Trained on original labels $\mathbf{u}_i$.
   • $\hat{G}_\epsilon$: Trained on perturbed labels $\tilde{\mathbf{u}}_i = \mathbf{u}_i + \boldsymbol{\epsilon}_i$, where $\boldsymbol{\epsilon}_i \sim \mathcal{N}(0, \sigma_\epsilon^2)$.
2. Raw Disagreement: For a new input $\mathbf{a}$, the raw uncertainty measure at coordinate $k$ is the pointwise absolute difference:
   $$d(\mathbf{a})_k = | \hat{G}(\mathbf{a})_k - \hat{G}_\epsilon(\mathbf{a})_k |$$
3. Stabilization: To prevent numerical instability (division by near-zero values) and ensure robustness, the raw disagreement undergoes two steps:
   • Spatial Smoothing: The disagreement map is smoothed using a $K \times K$ spatial window (e.g., $15 \times 15$) to average out local noise.
   • Floor Stabilization: A data-dependent floor $\tau_0$ is imposed, defined as a fraction of the median disagreement on the training set. The final scale is $\sigma(\mathbf{a})_k = \max(\bar{d}(\mathbf{a})_k, \tau_0)$.
4. Nonconformity Score: The conformal score for a calibration sample is defined as the maximum normalized residual over all output coordinates:
   $$S(\mathbf{a}, \mathbf{u}) = \max_k \frac{| u_k - \hat{G}(\mathbf{a})_k |}{\sigma(\mathbf{a})_k}$$
   This max-type score targets simultaneous coverage across the grid.

2.2 Calibration and Prediction

Using the split conformal protocol, the score $S$ is computed on a calibration set. The quantile $\hat{Q}_{1-\alpha}$ of these scores determines the width of the prediction bands. For a new input, the prediction band at coordinate $k$ is:
$$C(\mathbf{a})_k = \left[ \hat{G}(\mathbf{a})_k - \hat{Q}_{1-\alpha}\sigma(\mathbf{a})_k, \quad \hat{G}(\mathbf{a})_k + \hat{Q}_{1-\alpha}\sigma(\mathbf{a})_k \right]$$
Theoretical guarantees ensure that under exchangeability, the entire trajectory is simultaneously covered with probability $\geq 1-\alpha$.

3. Key Contributions

• Perturbation-Based Scale: Introduction of a method to derive adaptive uncertainty scales directly from the disagreement between a base model and a model trained on perturbed labels. This avoids the need for auxiliary uncertainty networks.
• Data Efficiency: The approach is specifically designed for the data-scarce regime. Unlike methods like UQNO that split the training data, the perturbation method utilizes the full proper training set for both the base and perturbed models, maximizing the information available for learning the operator.
• Simultaneous Coverage: The use of a max-type nonconformity score ensures that the prediction bands provide valid simultaneous coverage over the entire high-dimensional spatiotemporal output, rather than just marginal coverage at individual points.
• Stabilization Techniques: The inclusion of spatial smoothing and floor stabilization is shown to be critical for the efficiency of the conformal bands, preventing the inflation of uncertainty estimates due to isolated near-zero disagreements.

4. Experimental Results

Experiments were conducted on a 2D Navier–Stokes vorticity benchmark ($64 \times 64$ grid, 10 time steps) with a fixed total label budget of 1200 trajectories. The proposed method was compared against:

• MC Dropout: Using dropout at inference for variance estimation.
• Laplace Approximation: Diagonal last-layer approximation.
• UQNO: An auxiliary quantile network trained on a disjoint subset of data.
• Unscaled: A baseline with constant-width bands ($\sigma \equiv 1$).

Findings:

• Efficiency: The perturbation-based method produced substantially narrower conformal bands than all baselines while maintaining target simultaneous coverage.
  • At $\alpha=0.04$, the perturbation method achieved an average radius of 3.786, compared to 4.192 (Unscaled), 4.403 (Laplace), 8.530 (MC Dropout), and 16.784 (UQNO).
  • The improvement over UQNO was particularly significant (77.4% radius reduction), attributed to the data-splitting penalty inherent in training a separate uncertainty network.
• Ablation Studies:
  • Spatial Smoothing: Critical for performance. Removing smoothing increased the average radius from ~3.8 to ~83.6.
  • Floor Stabilization: Less critical for efficiency than smoothing but necessary for stability; removing it alone had a negligible effect on radius.
• Sensitivity: The method is robust to the magnitude of perturbation noise ($c \in [0.02, 0.10]$), with coverage remaining close to the nominal target across this range.
• Qualitative Analysis: The resulting uncertainty bands correctly widen in physically complex regions (vortex cores, shear layers) and contract in smoother laminar areas, aligning with the expected difficulty of the prediction task.

5. Significance and Claims

The paper claims that perturbation sensitivity serves as a practical and sample-efficient proxy for uncertainty in conformalized neural operators. The primary significance lies in decoupling uncertainty calibration from auxiliary modeling. By leveraging the disagreement between two models trained on nearly identical data, the method achieves adaptive, simultaneous coverage without sacrificing training data to a secondary network.

The authors position this as a general mechanism applicable to regression problems beyond PDEs, provided one can train two models on nearly identical data. The approach offers a data-efficient alternative for conformalized regression, particularly valuable when data is limited and the cost of training auxiliary models is prohibitive. The results suggest that for operator learning in the data-scarce regime, perturbation-based scaling outperforms existing adaptive methods (MC Dropout, Laplace, UQNO) in terms of band tightness while preserving rigorous statistical guarantees.