Learning Lattice Quantum Field Theories with Equivariant Continuous Flows

- The use of continuous flows to generate the field transformations,
preserving spatial symmetries
- The possibility of learning a family of models that parametrize
different values of the parameters present in the action.
- Better performance in the trainning.

1) Claims about better *scalability* of the trainning are not motivated (and probably not true).
2) Invertibility of their transformations at finite precision arithmetic has to be shown.

This paper present an important contribution to a very active área of
research in Lattice QCD: the possibility of using Machine Learning
(ML) techniques to samples of the Lattice Field Theories. The paper
presents some novel ingredients compared with previously published
works:

- The use of continuous flows to generate the field transformations,
preserving spatial symmetries
- The possibility of learning a family of models that parametrize
different values of the parameters present in the action.

I find these ingredients very interesting. The paper is well written
and in my opinion meets the criteria for publishing. Nevertheless
there are two points that I think that have to be either explained
better or corrected.

First, the paper insists that with their network architecture and
strategy (using as change of variables the solution of the ODE), the
scaling with the lattice size, measured using the ESS, improves
dramatically. But this point is very difficult to understand. Given
that the network architecture fully exploits invariance under
translations, and that the ESS is the exponential of an *extensive*
quantity (the difference of actions), it seems unavoidable that for
asymptotically large volumes, the ESS will deteriorate *exponentially
fast* at fixed training. I think that the authors have nicely shown
that their training is much more efficient than what has been found in
previous works. Still the numerical evidence (L/a=32 in two
dimensions) is really far from the interesting cases of the Lattice
QCD regime.

Second, there is a delicate numerical accuracy issue in their
construction. The type of flow equation that the authors use are most
probably parabolic in nature. It is well known that backwards
parabolic equations are not well-behaved. Numerical solutions may fail
to converge and/or the convergence might strongly depend on the
integrator used (it is also understood that high order Runge-Kutta
schemes are particularly bad). Note that this numerical instabilities
might arise for some values of the parameters that define the neural
network. For these reasons I think that it is important to include the
results of an experiment, where the integration forward in time is
performed, and then the integration backwards in time, and a measure
of the difference between the initial value of the parameters is
presented.

In summary I think that the statements about a better scaling might
need to be softened a bit (or more evidence needs to be
provided). Moreover some tests that invertibility of their
transformation is preserved at finite precision arithmetic has to be
presented.

On more general grounds, their approach share some ideas with the work
of M. Luscher "Trivializing maps, the Wilson flow and the HMC
algorithm". In the mentioned work the transformation, also based on a
very similar ODE, was constructed analytically instead of by numerical
methods, but still I think that a reference to this work should be
added to the text.

Once these two issues are addressed in the manuscript I will be very
happy to recommend the work for publication.

1) statements about a better scaling might
need to be softened a bit (or more evidence needs to be
provided)
2) tests that invertibility of their
transformation is preserved at finite precision arithmetic has to be
presented.
3) Reference seminal work by M. Luscher "Trivializing maps, the Wilson flow and the HMC algorithm".

1) Well-motivated
2) Of interest
3) Relatively original
4) Robust

1) Minor corrections are necessary

The submitted article presents a machine learning model to sample lattice field theories. The main goal of the work is to try and tackle the scalability challenges that still affect the sampling of lattice field theory configuration space using machine learning methods. The idea proposed herein is based on a continuous version of trivializing flow models (relying on a neural ordinary differential equation) and is tested in scalar field theory in two dimensions.
As compared to similar previous studies, the model presented here offers a better treatment of the symmetries of the theory (following ideas that have been previously applied for molecular modelling) and leads to an improved effective sample size even for relatively large lattices.
The work is well motivated and the topic is of interest. The idea combines existing methods in an original way and the numerical work appears to be robust. The manuscript is well written and the bibliography is quite complete.
The manuscript meets this journal´s acceptance criteria.
I request some corrections, which are included in the ´´Requested changes´´ list.

Requested corrections include the following:
1) On page 3, write that the explicit definition of the correlation length is given in the appendix.
2) On page 3, just before the beginning of Subsection 2.1, add some details about the CFTs that are expected to describe the critical point of the model, in particular depending on the degree of the potential.
3) Page 5: have different function bases H (among those that respect the symmetries of the problem and are sufficiently analytically tractable; for example, some families of polynomials or special functions?) been tested? If so, does the Fourier basis they use perform better?
4) Page 6: a couple of lines above Eq.(12): how is A defined?
5) Page 7: I would recommend replacing the word ´´experiment´´ with ´´numerical test´´ in the title of Section 4 and throughout the rest of the paper.
6) Page 9: second line: perhaps a word is missing in ´´by performing the training three runs´´? (Maybe the sentence was meant to be ´´by performing the training in three runs´´?)
7) Page 9: figure 4: the data appear to collapse on a common curve: have the authors tried to fit it to some known function?
8) Page 10, fifth line in Section 5: ´´even across the critical point´´: is it really so? Can the authors add more results or at least more comments on this?
9) Page 10, in Section 5, it would be interesting to know if the approach used in this work can be generalised to local symmetries, that is to gauge theories. It would be helpful if a discussion on this issue could be included in the Conclusions.
10) Page 11: last sentence of appendix A: replace ´´burn-in phase´´ with ´´thermalisation transient´´.
11) Page 14: add journal publication details for Reference 13: Phys.Rev.D 107 (2023) 5, 054501.
12) Page 14: in Reference 15 correct ´´Racaniere´´ with ´´Racanière´´.
13) Page 14: add journal publication details for Reference 20: PoS LATTICE2022 (2023) 036.
14) Page 15: add hyperlink for Reference 24.
15) Page 15: in Reference 31 correct ´´Noe´´ with ´´Noé´´.
16) Page 15: add hyperlink for Reference 32.