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Neural Network-Based Approach to Phase Space Integration

by Matthew D. Klimek, Maxim Perelstein

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Matthew Klimek
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-10-13
Date submitted: 2020-08-18 05:02
Submitted by: Klimek, Matthew
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Artificial Intelligence
  • High-Energy Physics - Phenomenology
Approach: Computational


Monte Carlo methods are widely used in particle physics to integrate and sample probability distributions (differential cross sections or decay rates) on multi-dimensional phase spaces. We present a Neural Network (NN) algorithm optimized to perform this task. The algorithm has been applied to several examples of direct relevance for particle physics, including situations with non-trivial features such as sharp resonances and soft/collinear enhancements. Excellent performance has been demonstrated in all examples, with the properly trained NN achieving unweighting efficiencies of between 30% and 75%. In contrast to traditional Monte Carlo algorithms such as VEGAS, the NN-based approach does not require that the phase space coordinates be aligned with resonant or other features in the cross section.

Published as SciPost Phys. 9, 053 (2020)

Author comments upon resubmission

We thank the referees for their helpful comments. Based on their feedback, and the editor's recommendation, we have incorporated a number of minor improvements to the draft, while the results and conclusions remain unchanged. Those improvements are listed here, and more detailed responses to the comments will be submitted below each referee report.

List of changes

- A typo above Eq. 1 was fixed.
- A new paragraph has been added starting below Eq. 3 discussing issues related to the bijectivity on the ANN.
- The discussion in the last paragraph of Sec. 3.1 has been modified to clarify that the ability to retrain quickly to nearby points in parameter space is not exclusive to the ANN algorithm.
- Fig. 5 now also shows the ANN output after unweighting, as validation that the unweighting procedure corrects any residual mismodeling. The last paragraph of Sec. 3.2 has been lengthened to describe the figure clearly.
- Grid lines have been added to Fig. 6, showing the result of applying VEGAS to this problem. The corresponding description of the plot in the caption and main text have been updated.
- There is a new Fig. 7 showing the raw output of the ANN and of VEGAS for the resonance that is not aligned with the phase space coordinate system. This is also described in the main text.
- The word "Appendix" has been added before references to the appendix.

Reports on this Submission

Anonymous Report 2 on 2020-9-5 (Invited Report)


1) Novel approach
2) Improved efficiency
3) Opens up a new research direction in the field


1) Implementation of ANN architecture


The authors have addressed the points in my previous report and the paper is now ready for publication in SciPost.

Requested changes


  • validity: high
  • significance: high
  • originality: top
  • clarity: top
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2020-9-2 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:1810.11509v3, delivered 2020-09-02, doi: 10.21468/SciPost.Report.1954


1) Novel method
2) Improves the efficiency for phase space integral, has the potential to break one of the bottleneck for particle generators
3) A thorough study of various 3-body structures


1) This study mainly shows improvements compared to non-NN bases methods, which is generically anticipated. It would be of much more importance and impact if the author explores which NN structure would be (quasi-)optimal for this important physics question of physics space integration for particle physics.
2) The author explores 3-body final states. It would be helpful to common on the expected scaling behavior once applied to a higher number of final state particles.


The authors performed a novel study using ANN to perform phase space integral, showing promising features when compared with more traditional methods. The study is robust and has great potential for MC generation applications.
The manuscript is of high quality and provides insights on an important topic. The authors have carefully addressed all the issues and suggestions raised in the previous reports. I recommend direct publication at SciPost.

  • validity: good
  • significance: good
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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