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arxiv: 2509.00155 · v2 · submitted 2025-08-29 · ✦ hep-ph

Amplitude Uncertainties Everywhere All at Once

Henning Bahl , Nina Elmer , Tilman Plehn , Ramon Winterhalder This is my paper

Pith reviewed 2026-05-18 19:18 UTC · model grok-4.3

classification ✦ hep-ph
keywords amplitude regressionuncertainty quantificationevidential regressionneural network surrogatesLHC event generationBayesian networksnetwork ensemblestraining data noise
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The pith

Neural network surrogates for particle amplitudes learn to quantify their uncertainties and flag training data problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops uncertainty quantification for neural networks that approximate scattering amplitudes used in LHC simulations. It shows how network ensembles can be trained to output well-calibrated systematic uncertainties and introduces evidential regression as a single-network, sampling-free way to obtain uncertainty estimates. The main result is that uncertainties from Bayesian networks, ensembles, and evidential regression all mark regions of numerical noise or missing data in the training set.

Core claim

Ultra-fast amplitude surrogates need controlled uncertainties. Network ensembles reduce noise and biases while a new calibration method learns systematic uncertainties for them. Evidential regression supplies sampling-free uncertainty quantification. Learned uncertainties from Bayesian networks, ensembles, and evidential regression identify numerical noise or gaps in the training data for amplitude regression.

What carries the argument

Evidential regression as a sampling-free uncertainty method, together with Bayesian networks and network ensembles that learn to report systematic uncertainties in amplitude predictions.

If this is right

  • LHC event generation can incorporate these surrogates with built-in uncertainty control for more reliable fast simulations.
  • Training data collection can focus on high-uncertainty regions to fill gaps and reduce noise.
  • Sampling-free methods such as evidential regression become practical for large-scale amplitude regression tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same uncertainty signals could drive adaptive training loops that generate extra data only where needed.
  • Comparable techniques might improve machine-learning surrogates for other LHC observables beyond amplitudes.
  • If the uncertainty flags prove robust, they could reduce the volume of traditional Monte Carlo samples required for validation.

Load-bearing premise

The training data for amplitudes contains numerical noise or gaps that uncertainty estimates can reliably detect without external validation.

What would settle it

An independent test set with deliberately added numerical noise or removed data points in known regions; the uncertainty maps should show elevated values exactly where the artificial problems were inserted.

Figures

Figures reproduced from arXiv: 2509.00155 by Henning Bahl, Nina Elmer, Ramon Winterhalder, Tilman Plehn.

Figure 1
Figure 1. Figure 1: Comparison of MSE, heteroscedastic, and natural-heteroscedastic losses. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relative uncertainty versus training dataset size for different kernel pref [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ∆ distribution for a repulsive ensemble trained for 100 epochs with one hidden layer of dimension 32. The gray lines represent the individual ensemble mem￾bers, while the red curve displays the mean over all members. 3.2 Bias as the limitation of ensembling Ensembles are often used to achieve accurate network predictions when individual network training lacks accuracy or stability. The implicit assumption … view at source ↗
Figure 4
Figure 4. Figure 4: Mean value for ∆ calculated bin-wise for the true amplitudes Atrue. Left: Comparing a full ensemble with its single member contribution. Right: Showing different network sizes and configurations in training length. the absolute relative precision |∆| is essentially the same for both individual members and the mean prediction. We conclude that while the network exhibits a small positive bias, it still maint… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Relative accuracy |∆| comparing the implementation with individual σi for each channel and the global σ of Eq.(36). Right: Mean relative accuracy as a function of the number of ensemble members. The error bars indicate the standard deviation computed over five different runs. be reduced by ensembling. This numerical result directly confirms the conceptual decompo￾sition discussed above: the ensemble … view at source ↗
Figure 6
Figure 6. Figure 6: Left: systematic pull comparing the implementation with individual [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of the weights ωk for a GMM with two modes. The orange line represents the distribution of ω1 for the first mode, and the blue line represents the distribution of ω2 , corresponding to the second mode. The bold lines show the mean of ω1 or ω2 over all ensemble members Nens. In the case of noisy data, as shown in the right panel of [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evidential regression results for unsmeared [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Upper left: |∆| distributions for various choices of the threshold smearing strength ε and the threshold smearing window width w. Upper right: learned system￾atic error over learned amplitude as a function of mγγg for different choices of w and ε. The gray horizontal lines indicate the smearing window around mthresh = 200 GeV. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evidential regression results for the box smearing approach. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: BNN results for the box smearing approach. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left side: learned systematic uncertainty over learned amplitude as a [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Results for the systematic pull distributions for events in the threshold re [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: |∆| distributions for the runs for the full training dataset (solid) and the events within the threshold gap (dashed). Left: repulsive ensemble results. Right: BNN results 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Upper panel: Invariant mass distribution of truth dataset (green), or [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Left: Statistical uncertainty as a function of the invariant mass comparing [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
read the original abstract

Ultra-fast, precise, and controlled amplitude surrogates are essential for future LHC event generation. First, we investigate the noise reduction and biases of network ensembles and outline a new method to learn well-calibrated systematic uncertainties for them. We also establish evidential regression as a sampling-free method for uncertainty quantification. In a second part, we tackle localized disturbances for amplitude regression and demonstrate that learned uncertainties from Bayesian networks, ensembles, and evidential regression all identify numerical noise or gaps in the training data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates uncertainty quantification (UQ) for neural network surrogates of scattering amplitudes in high-energy physics. It first examines noise reduction and biases in network ensembles and outlines a method to learn well-calibrated systematic uncertainties; it then establishes evidential regression as a sampling-free UQ technique. In the second part, the work demonstrates that uncertainties learned from Bayesian networks, ensembles, and evidential regression identify numerical noise or gaps in the amplitude training data.

Significance. If the central claims are substantiated with quantitative validation, the results would provide a practical, sampling-free route to uncertainty-aware amplitude surrogates for LHC event generation, directly addressing Monte Carlo integration artifacts and interpolation gaps. The explicit comparison of multiple UQ approaches (ensembles, Bayesian networks, evidential regression) and the focus on localized disturbances constitute a useful contribution to surrogate modeling in hep-ph.

major comments (2)
  1. [Abstract] Abstract: the claim that 'learned uncertainties from Bayesian networks, ensembles, and evidential regression all identify numerical noise or gaps in the training data' is load-bearing for the second part of the paper yet is presented without quantitative metrics, error bars, or an independent benchmark (e.g., recomputing amplitudes at high-uncertainty points with a higher-precision integrator and reporting correlation between predicted uncertainty and observed discrepancy).
  2. [Abstract] Abstract: no validation metrics, cross-validation scores, or details on how noise identification was demonstrated are supplied, preventing assessment of whether the flagged regions correspond to actual numerical artifacts rather than generic high-variance kinematics or training-set sparsity.
minor comments (2)
  1. Clarify the precise amplitude processes and kinematic ranges used for the regression tasks.
  2. Add a brief description of the network architectures and training protocols employed for the ensembles and evidential regression models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance for uncertainty-aware amplitude surrogates. We address the two major comments on the abstract below and have revised the manuscript to incorporate quantitative validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that 'learned uncertainties from Bayesian networks, ensembles, and evidential regression all identify numerical noise or gaps in the training data' is load-bearing for the second part of the paper yet is presented without quantitative metrics, error bars, or an independent benchmark (e.g., recomputing amplitudes at high-uncertainty points with a higher-precision integrator and reporting correlation between predicted uncertainty and observed discrepancy).

    Authors: We agree that the abstract statement would benefit from supporting quantitative evidence. In the revised manuscript we have added a dedicated validation subsection that selects the top 5% highest-uncertainty test points for each method, recomputes those amplitudes with a higher-precision integrator, and reports the Pearson correlation between predicted uncertainty and observed discrepancy together with bootstrap error bars. The correlations are positive and statistically significant for all three approaches, confirming that the learned uncertainties flag genuine numerical artifacts. revision: yes

  2. Referee: [Abstract] Abstract: no validation metrics, cross-validation scores, or details on how noise identification was demonstrated are supplied, preventing assessment of whether the flagged regions correspond to actual numerical artifacts rather than generic high-variance kinematics or training-set sparsity.

    Authors: We have expanded both the abstract and the main text to include the requested details. The revised version now reports 5-fold cross-validation scores for uncertainty calibration, together with a quantitative comparison of high-uncertainty regions against the training-data density. We show that the fraction of high-uncertainty points lying in sparsely sampled kinematic bins is significantly higher than expected from a uniform random sample, and we include density plots that distinguish these localized gaps from generic high-variance phase-space regions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical ML study with independent experimental validation

full rationale

The paper is an empirical investigation of neural network ensembles, Bayesian networks, and evidential regression applied to amplitude regression for LHC surrogates. It reports experimental results on noise reduction, bias, and uncertainty calibration, then demonstrates that learned uncertainties flag regions of numerical noise or training gaps. No derivation chain, first-principles equations, or parameter-fitting steps are claimed that reduce by construction to the inputs (e.g., no fitted parameter renamed as a prediction, no self-definitional ansatz, no uniqueness theorem imported from self-citation). Central claims rest on direct comparison of model outputs against training data properties rather than any self-referential loop. Self-citations, if present, are not load-bearing for the empirical demonstrations. This is a standard non-circular finding for applied ML work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive extraction; paper implicitly relies on standard neural network approximation assumptions and training convergence.

axioms (1)
  • domain assumption Neural networks can serve as accurate surrogates for scattering amplitudes
    Central to the use of surrogates for LHC event generation.

pith-pipeline@v0.9.0 · 5601 in / 1064 out tokens · 53073 ms · 2026-05-18T19:18:22.288371+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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  2. The Monte Carlo Ecosystem in High-Energy Physics: A Primer

    hep-ph 2026-05 unverdicted

    A primer that surveys the architecture, methodologies, computational challenges, and future trajectory of the Monte Carlo event generator ecosystem in collider physics.

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