Amplitude Uncertainties Everywhere All at Once

1. Problem Statement

In high-energy physics, specifically for the High-Luminosity Large Hadron Collider (HL-LHC), generating precise Monte Carlo event simulations is computationally expensive. Machine learning (ML) surrogates are increasingly used to approximate complex scattering amplitudes. However, for these surrogates to be reliable, they must not only predict the mean amplitude with high precision but also provide calibrated local uncertainty estimates.

The paper addresses three critical challenges:

1. Calibration of Systematic Uncertainties: Existing methods, particularly Repulsive Ensembles (REs), often fail to correctly calibrate systematic uncertainties in specific regions of phase space due to model biases.
2. Efficiency vs. Accuracy: Traditional Bayesian Neural Networks (BNNs) and Ensembles are computationally expensive. The authors investigate Evidential Regression (ER) as a sampling-free alternative.
3. Localized Data Deficiencies: Real-world amplitude calculations often suffer from numerical noise or gaps (e.g., near physical thresholds). The study tests whether ML models can identify and quantify these localized disturbances.

2. Methodology

The authors benchmark three probabilistic approaches for amplitude regression on the partonic process $gg \to \gamma\gamma g$:

A. Repulsive Ensembles (RE)

• Mechanism: Trains an ensemble of neural networks where a repulsive kernel term in the loss function prevents members from collapsing to the same minimum, thereby encoding statistical uncertainty via the spread of predictions.
• Bias Analysis: The authors investigate whether the ensemble mean reduces bias. They find that while ensembles reduce statistical noise, they do not eliminate systematic biases caused by limited model expressivity (e.g., insufficient network depth).
• Systematic Uncertainty Correction: They propose a novel two-step or combined training method where a separate network learns the systematic uncertainty ($\sigma_{syst}$) specifically for the ensemble mean, rather than averaging individual member uncertainties. This corrects miscalibration caused by the mismatch between ensemble variance and ensemble mean accuracy.

B. Evidential Regression (ER)

• Mechanism: Instead of sampling weights, ER places a prior (Normal-Inverse-Gamma distribution) over the likelihood parameters (mean and variance). The network outputs evidential parameters ($\gamma, v, \alpha, \beta$) which are analytically integrated to yield a Student-t predictive distribution.
• Advantage: This is a sampling-free method, making it computationally efficient.
• Degeneracy Handling: The authors compare two strategies to break the degeneracy in the four output parameters:
  1. Adding a regularization loss ($L_R$) penalizing high evidence for incorrect predictions.
  2. Constraining the relationship between parameters (e.g., $2\alpha = v$).
     They find the constraint method yields slightly better results.

C. Bayesian Neural Networks (BNN)

• Used as a benchmark (based on previous work) to compare against RE and ER, particularly in localized noise scenarios.

D. Localized Learning Challenges

The authors simulate realistic data issues by introducing:

1. Flat-box smearing: Gaussian noise applied to a specific invariant mass range.
2. Peaked smearing: Noise that increases sharply near a threshold.
3. Threshold gaps: Complete removal of training data in a specific phase-space region.

3. Key Contributions

1. Diagnosis of Ensemble Bias: The paper demonstrates that Repulsive Ensembles inherit the systematic bias of individual network members. If the network architecture lacks expressivity, the ensemble mean remains biased, and simple averaging of uncertainties leads to miscalibration.
2. Global Systematic Uncertainty Learning: A new method is proposed to learn $\sigma_{syst}$ directly for the ensemble mean. This resolves the miscalibration observed in REs, particularly for large ensemble sizes where the bias becomes the dominant error source.
3. Evidential Regression Validation: ER is established as a robust, efficient alternative to ensembles. The study shows that constraining evidential parameters ($2\alpha = v$) is superior to using regularization losses for this specific physics task.
4. Localization Capabilities: The study proves that all three methods (RE, ER, BNN) can successfully identify and quantify localized numerical noise and data gaps.
   • RE and BNN show superior ability to disentangle smooth underlying amplitudes from localized noise via interpolation.
   • ER performs well but struggles slightly with sharp edges in noise profiles compared to RE.
   • In data gap scenarios, RE and BNN correctly predict increased statistical uncertainty in the missing region, whereas ER tends to produce flat uncertainty estimates.

4. Key Results

• Uncertainty Calibration:
  • Standard REs miscalibrate systematic uncertainties when model bias is present. The proposed "global $\sigma_{syst}$" training fixes this, yielding well-calibrated pulls ($t_{syst} \sim \mathcal{N}(0,1)$) for small ensembles.
  • For large ensembles ($N_{ens} > 100$), residual non-Gaussian biases cause peaks in the pull distribution at $\pm 1$, indicating the Gaussian likelihood assumption breaks down for the bias component.
• Accuracy:
  • All methods achieve mean relative accuracy of $\langle |\Delta| \rangle \sim 3 \times 10^{-5}$ on clean data.
  • Ensemble accuracy scales as $1/\sqrt{N_{ens}}$ until hitting a "bias floor" determined by network expressivity.
• Localized Noise (Smearing):
  • Flat Box: All methods identify the smeared region. RE follows the expected noise profile most closely; ER and BNN provide better-calibrated pulls.
  • Peaked Threshold: RE and BNN successfully capture the sharp increase in noise near the threshold. ER struggles to capture the peak magnitude accurately.
• Data Gaps:
  • Networks maintain reasonable accuracy in gap regions due to interpolation capabilities (the amplitude is flat in the gap).
  • RE and BNN correctly predict a localized spike in statistical uncertainty within the gap.
  • BNN shows a global increase in uncertainty, while RE keeps the increase localized.
  • ER fails to capture the localized nature of the gap uncertainty, predicting a flat uncertainty profile.

5. Significance

This work provides a critical "toolkit" for the next generation of particle physics simulations:

• Reliability: It clarifies that simply using ensembles is insufficient for uncertainty quantification if the underlying model is biased; specific training strategies for systematic uncertainty are required.
• Efficiency: It validates Evidential Regression as a viable, fast alternative to ensembles for many scenarios, though it highlights limitations in handling sharp, localized data gaps.
• Robustness: The methods are proven to be robust against the types of numerical instabilities (threshold smearing, gaps) common in loop-induced amplitude calculations.
• Future Impact: These insights are essential for developing "surrogate" models that can replace traditional, slow calculations in High-Luminosity LHC event generators, ensuring that physics analyses remain statistically sound and systematic uncertainties are under control.