Original: Physical Benchmarking for AI-generated Cosmic web
Revised: Physical Benchmarking for AI-generated Cosmic web

Original: Abstract
Revised: Abstract

Original: The potential of deep learning based image-to-image translations has recently drawn a lot of attention; one intriguing possibility is that of generating cosmological predictions with a drastic reduction in computational cost.
Revised: The potential of deep learning based image-to-image translations has recently drawn a lot of attention; one intriguing possibility is that of generating cosmological predictions with a drastic reduction in computational cost.

Original: Such an effort requires optimization of neural networks with loss functions beyond low-order statistics like pixel-wise mean square error, and validation of results beyond simple visual comparisons and summary statistics.
Revised: Such an effort requires optimization of neural networks with loss functions beyond low-order statistics like pixel-wise mean square error, and validation of results beyond simple visual comparisons and summary statistics.

Original: In order to study learning-based cosmological mappings, we choose a tractable analytical prescription – the Zel’dovich approximation – modeled using U-Net, a convolutional image translation framework.
Revised: In order to study learning-based cosmological mappings, we choose a tractable analytical prescription – the Zel’dovich approximation – modeled using U-Net, a convolutional image translation framework.

Original: A comprehensive list of metrics is proposed, including higher-order correlation functions, conservation laws, topological indicators, dynamical robustness and statistical independence of density ﬁelds.
Revised: A comprehensive list of metrics is proposed, including higher-order correlation functions, conservation laws, topological indicators, dynamical robustness and statistical independence of density ﬁelds.

Original: We ﬁnd that the U-Net approach does well with some metrics but has difﬁculties with others.
Revised: We ﬁnd that the U-Net approach does well with some metrics but has difﬁculties with others.

Original: In addition to validating AI approaches using rigorous physical benchmarks, this study motivates advancements in domain-speciﬁc optimization schemes for scientiﬁc machine learning
Revised: In addition to validating AI approaches using rigorous physical benchmarks, this study motivates advancements in domain-speciﬁc optimization schemes for scientiﬁc machine learning

Original:  Introduction
Revised:  Introduction

Original: In the current era of ‘precision cosmology’, cosmological N-body simulations play a key role in the study of the large-scale structure of the universe, and are essential to interpreting the unprecedented amount of observational data available from sky surveys [4].
Revised: In the current era of ‘precision cosmology’, cosmological N-body simulations play a key role in the study of the large-scale structure of the universe, and are essential to interpreting the unprecedented amount of observational data available from sky surveys [4].

Original: In exploring cosmological parameter space, it is computationally challenging to evolve billions and trillions of particles at high enough resolution, even for a single simulation.
Revised: In exploring cosmological parameter space, it is computationally challenging to evolve billions and trillions of particles at high enough resolution, even for a single simulation.

Original: Since large numbers of simulations are needed for tasks such as covariance estimation [15], the computing requirements can easily become prohibitive.
Revised: Since large numbers of simulations are needed for tasks such as covariance estimation [15], the computing requirements can easily become prohibitive.

Original: Deep learning (DL) has exhibited great potential in learning highly complex functions and mappings, thereby attracting attention in various scientiﬁc domains, including cosmology [9].
Revised: Deep learning (DL) has exhibited great potential in learning highly complex functions and mappings, thereby attracting attention in various scientiﬁc domains, including cosmology [9].

Original: In particular, if DL methods can successfully capture the complexity of cosmic evolution, they can directly ad- dress the computational bottleneck, and even eliminate it in some cases.
Revised: In particular, if DL methods can successfully capture the complexity of cosmic evolution, they can directly ad- dress the computational bottleneck, and even eliminate it in some cases.

Original: However, issues such as interpretability, as well as the ability of DL models to make sufﬁciently accurate and physically meaningful predictions, are yet to be fully explored.
Revised: However, issues such as interpretability, as well as the ability of DL models to make sufﬁciently accurate and physically meaningful predictions, are yet to be fully explored.

Original: For example, can DL models, without any domain knowledge of cosmology, capture all the intricacies of nonlinear gravitational clustering?
Revised: For example, can DL models, without any domain knowledge of cosmology, capture all the intricacies of nonlinear gravitational clustering?

Original: And in what way can AI-based emulators complement standard numerical approaches?
Revised: And in what way can AI-based emulators complement standard numerical approaches?

Original: In this ongoing work, we model cosmic evolution using the widely adopted U-Net approach [5], applying it to a theoretically well-understood prescription, the Zel’dolvich approximation (ZA) [17].
Revised: In this ongoing work, we model cosmic evolution using the widely adopted U-Net approach [5], applying it to a theoretically well-understood prescription, the Zel’dolvich approximation (ZA) [17].

Original: Since the ZA is a simple dynamical mapping scheme, it has the beneﬁt of providing a clear physical picture while enabling a comprehensive study of the neural network’s physical interpretability.
Revised: Since the ZA is a simple dynamical mapping scheme, it has the beneﬁt of providing a clear physical picture while enabling a comprehensive study of the neural network’s physical interpretability.

Original:  Approximation for structure formation of the Universe
Revised:  Approximation for structure formation of the Universe

Original: Dynamical approximation : The Zel’dolvich approximation is the ﬁrst-order solution to the
Revised: Dynamical approximation : The Zel’dolvich approximation is the ﬁrst-order solution to the

Original: Lagrangian perturbation theory of structure formation.
Revised: Lagrangian perturbation theory of structure formation.

Original: It describes the motion of mass elements: an initially uniform distribution given by Lagrangian coordinates (cid:126)q is displaced by
Revised: It describes the motion of mass elements: an initially uniform distribution given by Lagrangian coordinates (cid:126)q is displaced by

Original: [Equation]
Revised: [Equation]

Original: ) where (cid:126)x ( r ) are the comoving coordinates, b ( t ) is the linear growth rate of ﬂuctuations and (cid:126)S ( (cid:126)q ) =
Revised: ) where (cid:126)x ( r ) are the comoving coordinates, b ( t ) is the linear growth rate of ﬂuctuations and (cid:126)S ( (cid:126)q ) =

Original: (cid:126) ∇ Φ( (cid:126)q ) is the gradient of the initial gravitational potential Φ( (cid:126)q ) .
Revised: (cid:126) ∇ Φ( (cid:126)q ) is the gradient of the initial gravitational potential Φ( (cid:126)q ) .

Original: The potential Φ( (cid:126)q ) is determined by the primordial density ﬂuctuations δ ( x ) via the Poisson equation, ∇ 2 Φ = 4 πGδ , where G is the
Revised: The potential Φ( (cid:126)q ) is determined by the primordial density ﬂuctuations δ ( x ) via the Poisson equation, ∇ 2 Φ = 4 πGδ , where G is the

Original: Newtonian constant of gravitation.
Revised: Newtonian constant of gravitation.

Original: Initial density perturbations are realizations of a Gaussian random ﬁeld, which is fully speciﬁed by a (given) power spectrum.
Revised: Initial density perturbations are realizations of a Gaussian random ﬁeld, which is fully speciﬁed by a (given) power spectrum.

Original: Uniformly placed particles representing mass elements are then moved according to the ZA.
Revised: Uniformly placed particles representing mass elements are then moved according to the ZA.

Original: Generation of the Dataset: For an initial power spectrum P ( k ) , the displacement ﬁeld S ( q ) is generated using FFT-based techniques, and the particles are transported from a regular grid via ZA displacements.
Revised: Generation of the Dataset: For an initial power spectrum P ( k ) , the displacement ﬁeld S ( q ) is generated using FFT-based techniques, and the particles are transported from a regular grid via ZA displacements.

Original: We generate 2000 pairs of ZA-evolved particles at two different time steps of the evolution, with respective scale factors a 0 = 0 .
Revised: We generate 2000 pairs of ZA-evolved particles at two different time steps of the evolution, with respective scale factors a 0 = 0 .

Original: 0464 (an “early” snapshot with redshift z 0 = 20 . 5 ) and a 1 = 0 .
Revised: 0464 (an “early” snapshot with redshift z 0 = 20 . 5 ) and a 1 = 0 .

Original: 215 (a “late” snapshot with redshift z 1 = 3 . 6 ).
Revised: 215 (a “late” snapshot with redshift z 1 = 3 . 6 ).

Original: The evolution of the so-called “cosmic web” in between these two moments is given by the ZA (from Equation 1), solely determined by the initial potential gradient.
Revised: The evolution of the so-called “cosmic web” in between these two moments is given by the ZA (from Equation 1), solely determined by the initial potential gradient.

Original: The displacement ﬁelds at these two times form the basis of the training/validation datasets.
Revised: The displacement ﬁelds at these two times form the basis of the training/validation datasets.

Original: Data is split such that 1000 pairs are used for training, 500 for validation and 500 for testing.
Revised: Data is split such that 1000 pairs are used for training, 500 for validation and 500 for testing.

Original: Each dataset is generated with a different random seed to ensure their statistical independence.
Revised: Each dataset is generated with a different random seed to ensure their statistical independence.

Original: The dataset consists of 64 3 N-body particles in a volume of 64 h − 1 Mpc and their displacements in three spatial dimensions.
Revised: The dataset consists of 64 3 N-body particles in a volume of 64 h − 1 Mpc and their displacements in three spatial dimensions.

Original: 2.1 Snapshot-to-snapshot translation using convolutional neural networks
Revised: 2.1 Snapshot-to-snapshot translation using convolutional neural networks

Original: A number of DL-based unsupervised generative models and supervised interpolation models have been applied in cosmological data creation [11, 7, 8].
Revised: A number of DL-based unsupervised generative models and supervised interpolation models have been applied in cosmological data creation [11, 7, 8].

Original: Except for a few applications where the loss functions are tailored to the physical problem, most of the applications are domain-agnostic.
Revised: Except for a few applications where the loss functions are tailored to the physical problem, most of the applications are domain-agnostic.

Original: That is, all the information about the underlying physics is entirely learned from the training data.
Revised: That is, all the information about the underlying physics is entirely learned from the training data.

Original: These data-driven models have shown good accuracy with validation datasets, albeit with respect to metrics that resemble the loss function.
Revised: These data-driven models have shown good accuracy with validation datasets, albeit with respect to metrics that resemble the loss function.

Original: [Figure]
Revised: [Figure]

Original: Architecture: We choose an image-to-image translation framework [12] originally used for biomedical images and recently applied in a cosmological setting [3, 5, 2, 1, 16].
Revised: Architecture: We choose an image-to-image translation framework [12] originally used for biomedical images and recently applied in a cosmological setting [3, 5, 2, 1, 16].

Original: We adopt the 3D U-Net architecture used in [5] with 15 convolution and deconvolution layers.
Revised: We adopt the 3D U-Net architecture used in [5] with 15 convolution and deconvolution layers.

Original: The architecture has a contracting path and then an expanding path.
Revised: The architecture has a contracting path and then an expanding path.

Original: A basic block consists of a convolution layer, batch normalization layer and a Rectiﬁed Linear Unit (ReLU) layer.
Revised: A basic block consists of a convolution layer, batch normalization layer and a Rectiﬁed Linear Unit (ReLU) layer.

Original: Mean squared error between the
Revised: Mean squared error between the

Original: Lagrangian co-ordinates is used as the loss function, with L2 regularization.
Revised: Lagrangian co-ordinates is used as the loss function, with L2 regularization.

Original: We refer the reader to
Revised: We refer the reader to

Original: Refs.
Revised: Refs.

Original: [12, 5] for further details of the network architecture.
Revised: [12, 5] for further details of the network architecture.

Original: Model Training: During the training of the 3D U-Net, a pair of initial-ﬁnal displacement ﬁelds is selected randomly from the dataset as the input, and the neural network is trained to learn the mapping between the z 0 and z 1 displacement ﬁelds.
Revised: Model Training: During the training of the 3D U-Net, a pair of initial-ﬁnal displacement ﬁelds is selected randomly from the dataset as the input, and the neural network is trained to learn the mapping between the z 0 and z 1 displacement ﬁelds.

Original: After the training is completed, the model is run on a set of test data, independent from training or validation, in order to compare model predictions with the ground truth (the ZA result).
Revised: After the training is completed, the model is run on a set of test data, independent from training or validation, in order to compare model predictions with the ground truth (the ZA result).

Original:  Validation Metrics
Revised:  Validation Metrics

Original: Once the loss has converged (after 10 training epochs), we ﬁrst visualize the reconstructed cosmic web.
Revised: Once the loss has converged (after 10 training epochs), we ﬁrst visualize the reconstructed cosmic web.

Original: Figure 1 shows the U-Net generated density (projected along the x -axis) of the cosmic density ﬁeld, demonstrating a reasonably good agreement with the ZA result.
Revised: Figure 1 shows the U-Net generated density (projected along the x -axis) of the cosmic density ﬁeld, demonstrating a reasonably good agreement with the ZA result.

Original: We next implement a number of more quantitative tests.
Revised: We next implement a number of more quantitative tests.

Original: In particular, we look at measures that quantify the matter distribution in the universe and topological connectivity of large scale structures.
Revised: In particular, we look at measures that quantify the matter distribution in the universe and topological connectivity of large scale structures.

Original: Details of these metrics are given below and summarized in Fig.
Revised: Details of these metrics are given below and summarized in Fig.

Original: Pixel-wise Comparison: The most straightforward comparison is the relative error in predicted densities or displacements.
Revised: Pixel-wise Comparison: The most straightforward comparison is the relative error in predicted densities or displacements.

Original: For a density ﬁeld ρ ( x ) , we deﬁne a relative error | ρ ( x ) pred − ρ ( x ) truth | /ρ
Revised: For a density ﬁeld ρ ( x ) , we deﬁne a relative error | ρ ( x ) pred − ρ ( x ) truth | /ρ

Original: ( x ) truth , where ρ ( x ) pred is the U-Net prediction, and ρ ( x ) truth is the ZA result.
Revised: ( x ) truth , where ρ ( x ) pred is the U-Net prediction, and ρ ( x ) truth is the ZA result.

Original: The last panel of Fig.
Revised: The last panel of Fig.

Original: 1 shows the relative difference for one of the realizations.
Revised: 1 shows the relative difference for one of the realizations.

Original: In addition, the one-point probability distribution function (PDF) is used to assess the prediction ﬁeld, which is also an important
Revised: In addition, the one-point probability distribution function (PDF) is used to assess the prediction ﬁeld, which is also an important

Original: [Figure]
Revised: [Figure]

Original: measure for nonlinear clustering in cosmic structure.
Revised: measure for nonlinear clustering in cosmic structure.

Original: The density PDF measurements of prediction and ground truth are shown in the ﬁrst panel of Fig.
Revised: The density PDF measurements of prediction and ground truth are shown in the ﬁrst panel of Fig.

Original: While relatively unbiased, we note that the density PDF error rises with density, which is concerning if the simulation is expected to capture correct clustering physics.
Revised: While relatively unbiased, we note that the density PDF error rises with density, which is concerning if the simulation is expected to capture correct clustering physics.

Original: Matter Power Spectrum: The power spectrum is the Fourier transform of the 2-point correlation function in real space, and is widely used as a statistical measure of the density ﬁeld.
Revised: Matter Power Spectrum: The power spectrum is the Fourier transform of the 2-point correlation function in real space, and is widely used as a statistical measure of the density ﬁeld.

Original: The matter overdensity δ ( x ) =
Revised: The matter overdensity δ ( x ) =

Original: ( ρ ( x ) − ρ ) /ρ , where ρ is the mean density.
Revised: ( ρ ( x ) − ρ ) /ρ , where ρ is the mean density.

Original: Writing its Fourier transform dual as δ ( k ) , the power spectrum P ( k ) is deﬁned by (cid:104) δ ( k ) δ ( k (cid:48) )
Revised: Writing its Fourier transform dual as δ ( k ) , the power spectrum P ( k ) is deﬁned by (cid:104) δ ( k ) δ ( k (cid:48) )

Original: (cid:105) =
Revised: (cid:105) =

Original: (2 π ) 3 P ( k ) δ 3 ( k − k (cid:48) ) .
Revised: (2 π ) 3 P ( k ) δ 3 ( k − k (cid:48) ) .

Original: Measurements of the predicted power spectra and ground truth P ( k ) are shown in the second panel of Fig.
Revised: Measurements of the predicted power spectra and ground truth P ( k ) are shown in the second panel of Fig.

Original: 2, where we observe differences of less than 5%, although there is a k -dependent bias in the error behavior.
Revised: 2, where we observe differences of less than 5%, although there is a k -dependent bias in the error behavior.

Original: Three-Point Correlation – Bispectrum:
Revised: Three-Point Correlation – Bispectrum:

Original: Apart from two-point correlation functions, higher-order statistics can also be used to characterise the density ﬁeld, and are especially useful for studying the late stages of structure formation where the evolution is highly nonlinear and non-Gaussian.
Revised: Apart from two-point correlation functions, higher-order statistics can also be used to characterise the density ﬁeld, and are especially useful for studying the late stages of structure formation where the evolution is highly nonlinear and non-Gaussian.

Original: Adapting similar symbol conventions as above, the bispectrum B ( k 1 , k 2 , k 3 )
Revised: Adapting similar symbol conventions as above, the bispectrum B ( k 1 , k 2 , k 3 )

Original: [6] is deﬁned as: (cid:104) δ ( k 1 ) δ ( k 2 ) δ ( k 3 ) (cid:105) =
Revised: [6] is deﬁned as: (cid:104) δ ( k 1 ) δ ( k 2 ) δ ( k 3 ) (cid:105) =

Original: (2 π ) 3 B ( k 1 , k 2 , k 3 )
Revised: (2 π ) 3 B ( k 1 , k 2 , k 3 )

Original: δ 3 ( k 1 + k 2 + k 3 ) .
Revised: δ 3 ( k 1 + k 2 + k 3 ) .

Original: Equilateral triangles in k space are selected for our bispectra analysis shown in the third panel of Fig.
Revised: Equilateral triangles in k space are selected for our bispectra analysis shown in the third panel of Fig.

Original: Deviation of the bispectra of the
Revised: Deviation of the bispectra of the

Original: U-Net generated ﬁelds with respect to ZA bispectra is considerably higher compared to the lower order metrics of the one-point PDF and matter power spectra.
Revised: U-Net generated ﬁelds with respect to ZA bispectra is considerably higher compared to the lower order metrics of the one-point PDF and matter power spectra.

Original: Topological metrics – Percolation Analysis: Topological statistics can be used to directly address the issue of connectivity of spatial structures, e.g., clusters of galaxies [14].
Revised: Topological metrics – Percolation Analysis: Topological statistics can be used to directly address the issue of connectivity of spatial structures, e.g., clusters of galaxies [14].

Original: For a given density threshold, the volume fraction of the excursion set f ES (i.e., the regions over the threshold) and the volume fraction of the largest region f 1 are computed.
Revised: For a given density threshold, the volume fraction of the excursion set f ES (i.e., the regions over the threshold) and the volume fraction of the largest region f 1 are computed.

Original: The behavior of f 1 /f ES as a function of f ES (last panel of Fig. 2) displays a rapid transition from zero to 1; this signiﬁes a topological shift from isolated clusters to a fully connected ﬁeld.
Revised: The behavior of f 1 /f ES as a function of f ES (last panel of Fig. 2) displays a rapid transition from zero to 1; this signiﬁes a topological shift from isolated clusters to a fully connected ﬁeld.

Original: For the input density ﬁeld ( z = z 0 ) , the percolation transition happens at a different density threshold than z = z 1 case, indicating a change in the topological behaviour.
Revised: For the input density ﬁeld ( z = z 0 ) , the percolation transition happens at a different density threshold than z = z 1 case, indicating a change in the topological behaviour.

Original: For this metric the NN results are very good and in excellent agreement with those from the ZA.
Revised: For this metric the NN results are very good and in excellent agreement with those from the ZA.

Original: Cross-power Test:
Revised: Cross-power Test:

Original: The accurate cosmic evolution follows ﬁxed dynamical rules, independent of the initial conditions.
Revised: The accurate cosmic evolution follows ﬁxed dynamical rules, independent of the initial conditions.

Original: However, the neural network emulating the evolution might show discernible bias inherited from the training dataset.
Revised: However, the neural network emulating the evolution might show discernible bias inherited from the training dataset.

Original: Speciﬁcally, we need to investigate whether the NN prediction induces otherwise non-existent correlations among the outputs, in contrast to the ZA-evolution of ﬁelds, where independent initial conditions should result in independent evolved ﬁelds.
Revised: Speciﬁcally, we need to investigate whether the NN prediction induces otherwise non-existent correlations among the outputs, in contrast to the ZA-evolution of ﬁelds, where independent initial conditions should result in independent evolved ﬁelds.

Original: An example test of a cross-power spectral measurement is shown in Fig. 3(b), where we generate two initial conditions, independent of the training set, measure their cross-power spectra, then evolve them separately by ZA and NN, and then measure the output cross-power between them again (Fig. 3(a) illustrates this validation route).
Revised: An example test of a cross-power spectral measurement is shown in Fig. 3(b), where we generate two initial conditions, independent of the training set, measure their cross-power spectra, then evolve them separately by ZA and NN, and then measure the output cross-power between them again (Fig. 3(a) illustrates this validation route).

Original: From Fig. 3(b), the cross-power shows that, at least for 2-point statistics, there are no hidden correlations among the NN-predicted density ﬁelds.
Revised: From Fig. 3(b), the cross-power shows that, at least for 2-point statistics, there are no hidden correlations among the NN-predicted density ﬁelds.

Original:  Discussion
Revised:  Discussion

Original: While DL-based translation networks and generative models have demonstrated signiﬁcant potential, critical tests have to be made before they are applied in high precision scientiﬁc ﬁelds like cosmology (where we are interested in error metrics of better than 1% [13]).
Revised: While DL-based translation networks and generative models have demonstrated signiﬁcant potential, critical tests have to be made before they are applied in high precision scientiﬁc ﬁelds like cosmology (where we are interested in error metrics of better than 1% [13]).

Original: In this work, we have demonstrated a subset of the validation schemes to motivate the scientiﬁc machine learning community towards using comprehensive sets of benchmarking tools.
Revised: In this work, we have demonstrated a subset of the validation schemes to motivate the scientiﬁc machine learning community towards using comprehensive sets of benchmarking tools.

Original: Starting from a tractable approximation to the complex dynamics of cosmic structure formation, we demonstrate the use of metrics that were not part of the optimization of the neural network.
Revised: Starting from a tractable approximation to the complex dynamics of cosmic structure formation, we demonstrate the use of metrics that were not part of the optimization of the neural network.

Original: For instance, topological transitions and bispectra are higher-order emergent features, agreement with which is not trivial in
Revised: For instance, topological transitions and bispectra are higher-order emergent features, agreement with which is not trivial in

Original: NN-based structure formation models.
Revised: NN-based structure formation models.

Original: These checks not only serve as robust validation tools, but also provide a clear direction for designing future cosmic structure emulation schemes.
Revised: These checks not only serve as robust validation tools, but also provide a clear direction for designing future cosmic structure emulation schemes.

Original: We note that although the DL results appear to be promising and close to the true physical results, in order for neural networks to compete with simulations, eventually all errors must be less than 1% for clustering metrics, and the spatial dynamic range of the 3D NN results must be signiﬁcantly extended, from a part in a hundred to parts per million.
Revised: We note that although the DL results appear to be promising and close to the true physical results, in order for neural networks to compete with simulations, eventually all errors must be less than 1% for clustering metrics, and the spatial dynamic range of the 3D NN results must be signiﬁcantly extended, from a part in a hundred to parts per million.

Original: An interesting and long road lies ahead.
Revised: An interesting and long road lies ahead.

Original:  Acknowledgements
Revised:  Acknowledgements

Original: The authors would like to thank ... for useful discussions.
Revised: The authors would like to thank Ben Gutierrez for useful discussions.

Original: The work is funded by ...
Revised: Work at Argonne National Laboratory was supported by the U.S. Department of Energy, Ofﬁce of High Energy Physics. Argonne, a U.S. Department of Energy Ofﬁce of Science Laboratory, is operated by UChicago Argonne LLC under contract no. DE-AC02- 06CH11357. This material is also based upon work supported by the

Revised: U.S. Department of Energy, Ofﬁce of Science, Ofﬁce of Advanced Scientiﬁc Computing Research and Ofﬁce of High Energy Physics, Scientiﬁc Discovery through Advanced Computing (SciDAC) program.

Original: TheZel’dolvich training data set are generated on Roomba, and the training is carried out on Swing, a system from the Laboratory Computing Resource Center (LCRC) of Argonne National Laboratory.
Revised: The training is carried out on Swing, a system from the Laboratory Computing Resource Center (LCRC) of Argonne National Laboratory.

Original: A Additional metrics
Revised: A Additional metrics

Original: The metrics studied in this analysis are by no means exhaustive.
Revised: The metrics studied in this analysis are by no means exhaustive.

Original: We list a few other metrics that will be incorporated into our validation scheme in the near future
Revised: We list a few other metrics that will be incorporated into our validation scheme in the near future.

Original: Minkowski Functionals: Minkowski Functionals (MFs) are topological descriptors of random ﬁelds that are useful in describing their statistical properties [10].
Revised: Minkowski Functionals: Minkowski Functionals (MFs) are topological descriptors of random ﬁelds that are useful in describing their statistical properties [10].

Original: Given a random ﬁeld ρ ( (cid:126)x ) , and its variance < ρ 2 > =
Revised: Given a random ﬁeld ρ ( (cid:126)x ) , and its variance < ρ 2 > =

Original: σ 20 , we can consider its excursion sets Σ( ν ) = x > νσ 0 which consists of all points that exceed the threshold νσ 0 .
Revised: σ 20 , we can consider its excursion sets Σ( ν ) = x > νσ 0 which consists of all points that exceed the threshold νσ 0 .

Original: Three MFs measure the area, length of boundary and genus that is characteristic of excursion sets:
Revised: Three MFs measure the area, length of boundary and genus that is characteristic of excursion sets:

Original: [Equation]
Revised: [Equation]

Original: where A is the area of the map, da and dl the area and boundary length elements, and K is the curvature of the boundary.
Revised: where A is the area of the map, da and dl the area and boundary length elements, and K is the curvature of the boundary.

Original: Dynamical Robustness: Apart from the density ﬁeld, the matter velocity ﬁeld is of equal signiﬁcance regarding the correct description of the evolution of the cosmic web, and velocity-density cross correlations are crucial for this measurement.
Revised: Dynamical Robustness: Apart from the density ﬁeld, the matter velocity ﬁeld is of equal signiﬁcance regarding the correct description of the evolution of the cosmic web, and velocity-density cross correlations are crucial for this measurement.

Original: For ZA evolution, explicit reversibility can be also checked to guarantee the dynamical robustness of reconstruction.
Revised: For ZA evolution, explicit reversibility can be also checked to guarantee the dynamical robustness of reconstruction.