Conformal prediction for uncertainties in nucleon-nucleon scattering

Conformal prediction is a distribution-free and model-agnostic uncertainty-quantification method that provides finite-sample prediction intervals with guaranteed coverage. In this work, for the first time, we apply conformal-prediction to generate uncertainty bands for physical observables in nuclear physics, such as the total cross section and nucleon-nucleon phase shifts. We demonstrate the method’s flexibility by considering three scenarios: (i) a pointwise model, where expansion coefficients in chiral effective field theory are treated as random variables; (ii) a Gaussian-process model for the coefficients; and (iii) phase shifts at various energies and partial waves calculated using local interactions from chiral effective field theory. In each case, conformal-prediction intervals are constructed and validated empirically. Our results show that conformal prediction provides reliable and adaptive uncertainty bands even in the presence of non-Gaussian behavior, such as skewness and heavy tails. These findings highlight conformal prediction as a robust and practical framework for quantifying theoretical uncertainties.

💡 Research Summary

This paper presents the first application of conformal prediction (CP) to quantify theoretical uncertainties in nucleon‑nucleon (NN) scattering, a problem of central importance in low‑energy nuclear physics. The authors begin by motivating the need for robust uncertainty quantification (UQ) in the era of high‑precision ab‑initio calculations and chiral effective field theory (χEFT). While Bayesian methods and Gaussian‑process (GP) models have become standard tools for propagating EFT truncation errors and fitting low‑energy constants, they rely on prior choices, model specifications, and often assume Gaussian residuals. CP, by contrast, is a distribution‑free, model‑agnostic framework that guarantees finite‑sample marginal coverage under the sole assumption of exchangeability (i.i.d. data).

The methodological core of the work is a clear exposition of how quantile regression can be combined with split‑conformal techniques to produce prediction intervals that adapt to heteroscedasticity and non‑Gaussian features. The authors describe the standard split‑conformal pipeline: (i) train a predictive model on a training set, (ii) compute conformity scores (absolute residuals) on a calibration set, (iii) select the (1‑α)‑quantile of these scores, and (iv) form symmetric intervals around the model prediction. They also discuss the conformalized quantile regression (CQR) variant, which yields asymmetric intervals by adjusting the conditional quantile estimates with conformity scores.

Three distinct case studies illustrate the flexibility of CP.

1. Point‑wise χEFT model: Expansion coefficients of the χEFT series are treated as independent random variables. By sampling these coefficients and computing total cross sections, the authors construct CP intervals that widen for higher‑order terms where coefficient variability is larger, thereby reflecting the physical intuition of increasing truncation error.
2. Gaussian‑process model: Here the coefficients are modeled as smooth functions of the laboratory energy using a GP. While the GP provides mean and variance predictions, the raw GP credible intervals often under‑cover. Applying CP to the GP outputs (using the GP mean as the point prediction) yields calibrated intervals with empirical coverage close to the nominal 95 % level.
3. Phase‑shift calculations from a realistic local χEFT Hamiltonian: Phase shifts for several partial waves (e.g., ^1S_0, ^3P_2) are computed across a range of energies. The resulting residual distributions exhibit skewness and heavy tails, especially near resonant energies. CP successfully generates non‑symmetric bands that adapt to these features, and extensive validation shows that the empirical coverage remains at the target level for each energy and partial wave.

Across all scenarios, the authors report empirical coverages of 0.945–0.950 for nominal 95 % intervals, confirming the theoretical guarantees of CP even when the underlying data are far from Gaussian. They also emphasize that CP naturally flags model misspecification: poorly fitted models produce wide intervals, whereas well‑calibrated models yield tighter bands.

The discussion acknowledges that CP’s exchangeability assumption can be violated in time‑dependent or non‑stationary data. The paper outlines possible extensions—weighted conformity scores, non‑symmetric algorithms, and recent advances in high‑dimensional CP—to handle such cases. Moreover, the authors propose a hybrid workflow where CP is applied as a post‑processing step to Bayesian posteriors, thereby marrying the interpretability of Bayesian credible intervals with the finite‑sample frequentist guarantees of CP.

In summary, this work demonstrates that conformal prediction offers a powerful, assumption‑light alternative for uncertainty quantification in nuclear theory. It delivers calibrated, adaptive prediction bands for cross sections and phase shifts, remains robust under skewed and heavy‑tailed error structures, and provides a diagnostic tool for model adequacy. The study opens the door for broader adoption of CP in other areas of theoretical physics where reliable error bars are essential for meaningful comparison with experimental data.