Learning the Simplicity of Scattering Amplitudes

The article presented by the authors provides a novel and compelling
approach to simplifying expressions in spinor-helicity variables,
addressing challenges posed by redundancies from momentum conservation
and Schouten identities. By leveraging machine learning (ML)
techniques, the authors provide a fresh perspective on this complex
problem. The presentation is clear and detailed. First, the authors
present a one-shot simplification technique for expressions of
moderate size. Then, they expand on this by investigating a sequential
simplification approach where sub-expressions are simplified after
being grouped together based on their cosine similarity. This allows
the simplification of larger expressions. Their open-source code
available on GitHub adds further value to their contribution.

In summary, I believe this article will be a valuable contribution to
both the machine learning and the scattering amplitude literature. I
appreciate the effort made by the authors to connect these fields
while making the article accessible to both communities. I therefore
recommend its publication.

Beforehand, I suggest the following minor revisions to enhance clarity
and impact:

1. Towards the end of page 2, in the introduction, the authors review
ML applications to high-energy physics. It may be worthwhile to
include in this discussion prior studies aimed at reproducing the
numerical output of the amplitudes, rather than achieving exact
analytic simplifications. For example, consider referencing
arXiv:2002.07516 and arXiv:2107.06625, (and perhaps check
references therein). Mentioning these works would strengthen the
connection to existing literature.

2. Towards the end of page 5, the authors cite references [40-42] in
the context of little-group and dimensional analysis
constraints. While these are indeed relevant, the main
simplification in those works arises from analysing
singularities. Perhaps the wording could be rephrased as "These
constraints, together with information from singular kinematic
limits, can [...]" to more accurately reflect that
work. Additionally, arXiv:2203.04269 is a recent advancement in
this approach, which can simplify spinor-helicity expressions in
the roundabout way (complex analytic -> numeric -> simpler
analytic).

3. On page 9, in section 2.4, projective coordinates and twistors are
mentioned, but without any reference. In the context of momentum
twistors arXiv:0905.1473 comes to mind, and additional references
could help guide readers unfamiliar with these topics.

4. On page 11, the authors mention that a numerical check is performed
on candidate expressions generated by the one-shot simplification
approach to verify their validity. Looking in the code at
add_ons/numerical_evaluations.py, it appears they are using
double-precision (16-digit) phase space points, requiring 9-digit
precision to declare a value as zero (ZERO_ERROR_POW_LOCAL = 9). It
might be beneficial stating this in the paper.

In principle, one may be concerned that numerical similar, but not
analytically identical, simplified expressions could be erroneously
accepted, or, on the contrary, that valid simplifications could be
discarded due to precision loss. While this is probably unlikely
until expressions have hundreds or thousands of terms, it might be
worth commenting upon. Higher-precision and/or finite-field
evaluations would greatly reduce room for errors, if needed.

The authors may also wish to consider a native python
implementation of spinor-helicity evaluations, rather than using a
Mathematica link to S@M; the python package "lips" could be an
alternative.

5. The particular redundancy of four-point amplitudes is referred to
on multiple occasions. A more mathematically sound statement is
that at four point, factorization is not unique (see "unique
factorization domain", arXiv:2203.17170). While at n-point, n>4,
factorization is (conjecturally) unique. This implies that there
exists a unique least common denominator (LCD) for n>4, but not
n=4.

This is evident in the first two amplitudes in appendix G, which
admit representations with different denominators. The first
Parke-Taylor formula is more commonly written as
(⟨1|2⟩^3)/(⟨1|4⟩⟨2|3⟩⟨3|4⟩), while the second expression could be
written as (⟨1|2⟩^6[3|4])/(⟨1|3⟩⟨1|4⟩⟨2|3⟩⟨2|4⟩⟨3|4⟩). The authors
could comment on how this choice is made: does the ML model return
multiple candidate representations, and is one picked at random
among the valid ones? Or perhaps, are the denominator factors in
the simplified expression restricted to be a subset of those in the
original expression?

Similarly, for n>4, the authors could comment on the ability of the
model to identify the least common denominator. For instance, in
the last amplitude before the bibliography, the denominators
contain manifestly spurious factors [25] and [45]. I imagine this
is an artefact of the compact form chosen to write this expression
in the paper, even if the ML algorithm may return an expression
without those factors in the denominator. It is worth noting that a
clear and efficient algorithmic way to determine the LCD exists,
through univariate interpolation and factor matching.

6. On a related note to 5, it has been widely observed that since
rational functions in scattering amplitudes are sparse,
multivariate partial fraction decompositions are an important tool
to tame their complexity. The authors could comment on whether this
already affects their ML approach or how it could be included.

7. While the authors consider up to six-point amplitudes, it appears
that only maximally helicity violating trees are considered. It
might be worthwhile to comment on what changes would be required to
handle NMHV trees, which may include three-particle irreducible
denominator factors s_ijk, and potentially spurious spinor chains.

Similarly, I would imagine that a more compelling application of
this method would be to loop-integral rational coefficients, rather
than tree amplitudes. Like for NMHV trees, these may include
several more denominator factors, other than 2-particle spinor
brackets.

8. In loop amplitudes, the numerical constants in the rational
coefficients can be rational numbers with fairly large denominators
and numerators. In their work, the authors encounter mostly simple
numerical coefficients (like ±1 or ±2), and by default choose to
blind all constants (see page 22). They could comment on how their
method could be reconciled with the numbers observed in loop
coefficients. Perhaps a similarity could be defined among the
constants, on top of that among the spinor monomials?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

The manuscript presents a machine learning approach for the simplification of scattering amplitudes, more precisely for the algebraic simplification of rational functions in spinor helicity variables. Such a simplification is indeed a challenge and one that many scientists in this field have stumbled upon.

The authors start by discussing the type of input and output amplitudes/expressions and the generation of the training data set via a backward generation. Two approaches, a one-shot simplification and a sequential version, are then discussed in depth. Both setups -- including transformer models, beam searches, and contrastive learning -- consist of state-of-the-art ML methods. The network architectures are carefully chosen for the problem at hand and are reasonably motivated by the authors.

For both approaches the manuscript includes a plethora of results for the performance that show high efficacy and success rates for many applications, but also point to limitations of these methods. I agree with the authors' conclusion that their approach has the power and flexibility to tackle real problems in theoretical high-energy physics (and beyond!).

The article is written in a clear and concise language and understandable, in most parts, also for non-experts in machine learning. I highly appreciate the authors' efforts in that respect, as the target group -- mostly theoretical physicists -- might not be very familiar with many of ML concepts.

Furthermore, the methods described here are a beautiful showcase of the application of ML to obtain exact results. I believe that these and similar methods are also applicable to other areas of theoretical physics where one seeks exact analytical data.

Finally, the submission is accompanied by a git repository containing the code and data set used in this project, which allows for reproductability of the results and which is a valuable resource for the community.

I, therefore, recommend this article for publication after a few minor things have been answered/corrected in an updated version:
1 - In Sec 2.2: I would like to see some justification for the choice of the target data set and/or how well it will work for more complicated terms and/or how is there any bias by doing a backward generation compared to a forward generation (see e.g. https://arxiv.org/pdf/1912.01412). It seems biased to me in the sense that it might work great for amplitudes where we expect such very simple final forms, but I wouldn't expect it to capture well more complicated final expressions. I am thinking here of amplitudes with a "simplest" form of 10-20 numerator terms or even an application of these ideas to intermediate expressions where one might be interested to simplify very complicated rational functions to slightly less complicated rational functions. Or formulated from a different perspective: Is the target data set optimal for the sequential simplification or is there some potential for improvement?
2 - In the end of section 2.2 the authors mention that their setup is restricted to unitary relative coefficients of terms. This seems quite a bit restrictive for "real-world" applications (e.g. not N=4 SYM). I have seen amplitudes where the (probably) "simplest" form contains relative coefficients like 3/7. There seems to be a partial resolution to this restriction in footnote 14, but I'd suggest to extend that discussion as it wasn't clear to me how this additional difficulty could be handled most efficiently.
3 - Similar to the point above, the restriction to Nterms <= 3 in section 2.2 might not be optimal for more complicated problems. In summary, some more care should be taken to the choice of data sets and a discussion of their validity/bias/potential for improvement would be useful to the reader.
4 - How does the approach described in this paper compare to existing software, as e.g. cited in [13,14]?

Further ideas for improvement:
5 - Section 2.4 seems out of place and most of its content might better fit into the introduction.
6 - The same holds for the last few sentences of section 4.1. These comments might fit better in thes introductory part of section 4 or even in the introduction section.
7 - Figure 12: It might be useful to also somehow mark the distinction of circle and triangle markers within the plots.
8 - It would be interesting to see a comparison of the approach described in this paper with existing software, as e.g. cited in [13,14].
9 - The conclusions might benefit from an extended discussion related to future developments in this research area. In particular it might be worth emphasizing that these methods produce exact analytical results and may be applicable to other problems where fast numerical checks of the answers are available. I can easily think of a handful of applications that fulfill the latter criterion.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

In the present article, the authors consider the use of machine-learning techniques to simplify tree-level scattering amplitudes written in terms of spinor-helicity variables, a task of relevance to analytic calculations in theoretical high-energy physics.

While machine-learning techniques have in the past years revolutionized many different fields related to numeric and noisy data, comparably little work has been done on applying machine-learning techniques to analytic and exact subjects, such as theoretical physics and mathematics. The present paper is a very interesting specimen of such work. As such, it indeed provides a novel link between different research areas and opens a new pathway in an existing research field, with clear potential for multi-pronged follow-up work.

In an introduction, the authors summaries the challenge and prospective reward behind simplifying complicated expression in theoretical physics, and in particular those written in terms on spinor-helicity variables.
They sketch their machine-learning approach to this challenge and give an overview of related machine-learning approaches in the literature.

In the second section, the authors introduce the spinor-helicity formalism and describe how they construct their training data by applying a number a scrambling steps to randomly generated simple expressions to produce expressions that should be simplified in a supervised-learning approach.

In the third section, the authors describe a transformer model that is able to simplify an expression if the expression is related to the simple expression by three or less scrambling steps. However, the authors find that the accuracy drops with the number of scrambling steps and does not generalize beyond the number of steps seen at training.

In a fourth section, the authors address the problem in generalization by training a second transformer to identify subsets of terms that are likely to be simplified by the first transformer. This allows the combined model to reliably simplify expressions of up to 50 terms. They also demonstrate that their model is able to simplify complicated expressions arising from tree-level scattering amplitudes.

The authors conclude with a summary of their results and perspective on future work. In particular, the authors argue that a similar machine-learning approach could be used for a range of different simplification problems.

The authors give further more technical details on their approach in seven appendices. They have made their code available via a github repository, further facilitating the reproduction of their work.
Moreover, the authors provide a link to an interactive online demonstration of their model, allowing interested readers to apply the model and gauge its strengths and limitations themselves.

The paper is well written and presents very interesting results. I thus recommend it for publication provided the following minor comments are addressed in a satisfactory fashion:

1. The authors mention their interactive online demonstration only in the abstract. It might improve the impact of the paper to refer to this interactive demonstration also in other places, as well as to elaborate on its capabilities.

2. One case for machine-learning in this context is that there exists no clear algorithmic way to simplify expressions in spinor-helicity variables analytically. The authors make this case in section 2.4, but this seems to be so fundamental to their work that it might already be mentioned in the introduction.

3. A second case for machine-learning in this context is that the simplification is hard to achieve but that its correctness is easy to verify via numerics. The authors mention this numeric verification on page 11. But again, this aspect seems to be so fundamental to their work that it would benefit from being mentioned already in the introduction. (It is widely known that transformers are prone to hallucinations. The possibility to numerically verify their output is the reason that hallucinations are not a problem for the authors' approach.)

4. One of the motivations that the authors bring up for their work is the simplicity of the Parke-Taylor amplitude (1.1). The authors mention that their model successfully simplifies the corresponding expressions for four and five gluons. I could not find a corresponding statement for six gluons though. In contrast, Parke and Taylor successfully simplified their earlier results for six gluons in 1986, using slightly different but related variables. (Parke and Taylor came up with an educated guess that they checked numerically, but that is also how the author's model works.) Could the authors place their impressive(!) achievements using a machine-learning approach into perspective of what is currently happening in the field of scattering amplitudes using a more traditional approach?

5. On page 11, the authors give many relevant technical details on the training of their model. Could they also mention how long training took on the A100 GPUs that they used?

6. From figure 3, it seems like five-point amplitudes are harder to learn for the model than four- and six-point amplitudes. Do the authors have an explanation as to why?

7. In figure 8, the authors give the averages of cosine similarities. Would it be useful to give also standard deviations?

8. Below (4.6), the authors write ``even as c(t) increases''. Since $0<c_0<1$ and $\alpha>0$, doesn't c(t) decrease with t?

9. While the authors consider massless amplitudes, many interesting processes involve also massive particles. Could the authors comment on whether it is possible to extend their approach to the massive case, for which a variant of spinor-helicity variables exists as well?

10. As previously written, the paper is in general very well written. I have spotted only two typos that the authors might wish to correct. On page 25, ``structure words'' should likely read ``structure of words''. On page 31, ``away [...] form completely dissimilar terms'' should likely read ``away [...] from completely dissimilar terms''.

Ask for minor revision