Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modeling

This work introduces a method to equip data-driven polynomial chaos expansion surrogate models with intervals that quantify the predictive uncertainty of the surrogate. To that end, jackknife-based conformal prediction is integrated into regression-based polynomial chaos expansions. The jackknife algorithm uses leave-one-out residuals to generate predictive intervals around the predictions of the polynomial chaos surrogate. The jackknife+ extension additionally requires leave-one-out model predictions. Both methods allow to use the entire dataset for model training and do not require a hold-out dataset for prediction interval calibration. The key to efficient implementation is to leverage the linearity of the polynomial chaos regression model, so that leave-one-out residuals and, if necessary, leave-one-out model predictions can be computed with analytical, closed-form expressions. This eliminates the need for repeated model re-training. The conformalized polynomial chaos expansion method is first validated on four benchmark models and then applied to two electrical engineering design use-cases. The method produces predictive intervals that provide the target coverages, even for low-accuracy models trained with small datasets. At the same time, training data availability plays a crucial role in improving the empirical coverage and narrowing the predictive interval, as well as in reducing their variability over different training datasets.

💡 Research Summary

This paper introduces a novel framework that equips data‑driven polynomial chaos expansion (PCE) surrogate models with statistically valid prediction intervals, thereby providing uncertainty quantification for deterministic regression surrogates. The core idea is to combine regression‑based PCE with jackknife and jackknife+ conformal prediction (CP) methods, exploiting the linear nature of ordinary‑least‑squares (OLS) PCE to compute leave‑one‑out (LOO) residuals and LOO predictions analytically, without any model retraining.

Methodology
A PCE surrogate approximates a scalar output y = μ(x) as a finite sum of orthonormal multivariate polynomials Ψ_k(x) with coefficients c_k obtained by solving a linear least‑squares problem. The design matrix D (size M×K) contains evaluations of the basis functions on the M training points. Because the regression is linear, the hat matrix H = D(DᵀD)⁻¹Dᵀ can be formed directly. The LOO residual for the m‑th training point is then r_LOO,m = (y_m – ŷ(x_m)) / (1 – h_mm), where h_mm is the m‑th diagonal entry of H. For jackknife+ CP, the LOO prediction at a new input x* is ŷ^{‑m}(x*) = ŷ(x*) – (d_* (DᵀD)⁻¹ d_mᵀ)·r_LOO,m / (1 – h_mm), with d_* and d_m being the basis‑function vectors at x* and x_m, respectively.

Using absolute residuals as non‑conformity scores, the jackknife interval is symmetric around the full‑model prediction ŷ(x*), while the jackknife+ interval is asymmetric and built from the ensemble of LOO predictions. The jackknife+ interval enjoys a formal coverage guarantee of (1‑2s) for significance level s, whereas both methods achieve empirical coverage close to the target (1‑s).

Experiments
The authors validate the approach on four benchmark functions (varying dimensionality and sample sizes) and two engineering design problems: (1) a heat‑sink thermal management case and (2) a Stern‑Gerlach electromagnet design. For each case, they assess empirical coverage, average interval width, and variability across multiple random training sets, targeting 90 % and 95 % coverage levels.

Results

• Empirical coverage matches the target across all experiments; with larger training sets (M ≥ 100) the achieved coverage is within 0.2 % of the nominal level.
• Interval widths shrink markedly as the number of training points grows, reflecting improved PCE accuracy and smaller LOO residuals. With very small data (M ≤ 30) intervals are wider but still maintain proper coverage, indicating robustness to limited data.
• Jackknife and jackknife+ produce virtually identical intervals in practice, and the analytical LOO computation eliminates the O(M) model‑retraining cost that plagues standard cross‑CP methods.
• A brief exploration of normalized non‑conformity scores shows potential for handling heteroscedastic noise, suggesting a path toward adaptive conformal intervals.

Contributions and Significance

1. First integration of PCE with jackknife‑based conformal prediction, yielding a “conformalized PCE” that provides distribution‑free, finite‑sample uncertainty quantification without sacrificing data efficiency.
2. Derivation of closed‑form LOO residuals and predictions for OLS‑PCE, removing the computational bottleneck of traditional cross‑validation‑based CP.
3. Demonstration that reliable prediction intervals can be obtained even in small‑data regimes typical of expensive engineering simulations, facilitating uncertainty‑aware design and optimization.
4. Discussion of extensions to multivariate outputs, sparse or adaptive polynomial bases, and heteroscedasticity‑aware normalization, opening avenues for broader application in scientific computing.

Overall, the paper presents a practical, theoretically sound, and computationally cheap solution for equipping deterministic surrogate models with calibrated uncertainty estimates, bridging a gap between classical uncertainty quantification methods and modern data‑driven surrogate modeling.