AI-Powered Reconstruction of Dark Matter Velocity Fields from Redshift-Space Halo Distribution

In order to train and validate our deep learning model, we employed the CosmicGrowth simulations Jing (2018) as our training and test datasets. The CosmicGrowth simulation is a valuable tool for investigating the growth and evolution of DM halos and subhalos in the universe. The simulation was conducted in a box with a side length of 1.2 $h^{-1}{\rm Gpc}$, containing $2048^{3}$ DM particles with a mass resolution of $M_{\rm DM}=3.8\times 10^{9}~{}h^{-1}M_{\odot}$.

In this study, we adopt the standard flat $\Lambda$CDM cosmological model, which is well consistent with the results of WMAP cosmology Komatsu et al. (2011); Hinshaw et al. (2013). The standard cosmological parameters adopted are as follows: $\{\Omega_{c},\Omega_{b},h,n_{s},\sigma_{8}\}=\{0.2235,0.0445,0.71,0.968,0.83\}$.

In order to construct the halo/subhalo catalog in redshift space at $z=0.59$, it is necessary to take the number density of the input data to be $3\times 10^{-3}~{}{\rm Mpc}^{-3}h^{3}$. This value is consistent with current spectroscopic observations. The relationship between the redshift space position, denoted by $\bm{s}$, and the real space position, denoted by $\bm{r}$, after accounting for the RSD effect is given by

where $\bm{v}$ denotes the peculiar velocity, $a$ denotes the scale factor, $H(a)$ denotes the Hubble parameter, and $\hat{z}$ represents the unit vector along the line of sight (LoS). The density and velocity fields were computed based on the catalogue samples, with the haloes assigned to a $512^{3}$ mesh using the CIC (Cloud-in-Cell) scheme, with a cell resolution of $2.35h^{-1}~{}{\rm Mpc}^{3}$.

The objective is to construct a mapping based on the UNet method, which will take as its input a sparse halo number density field in redshift space (to provide an accurate representation of real observations) and output the real-space fields, including the DM density field and DM velocity/momentum fields.

For clarification, let us first define two density fields that will be used during the UNet reconstruction process: 1) the redshift-space sparse halo number density field, $\rho_{s}$; 2) the real-space DM density field, $\rho_{\rm DM}$.

Moreover, the real-space comoving velocity field is denoted by $\bm{v}$, with its magnitude represented by $|\bm{v}|$ and its direction by $\hat{v}$. The momentum field in real space is defined as the number-weighted velocity, given by $\bm{m}=(1+\delta_{\rm DM})\cdot\bm{v}$, where $\delta_{\rm DM}\equiv\rho_{\rm DM}/\bar{\rho}_{\rm DM}-1$ is the DM density field relative to its mean.

The reconstruction process is divided into two principal steps, as illustrated in Fig. 1, which provides a detailed illustration of the UNet network in Fig. 2. For the illustrative purpose, the velocity will be taken as an example. The same methodology is employed for the processing of the momentum field.

In what follows, we will provide a comprehensive description of this process.

1) The mapping from $\rho_{s}(\bm{x})$ to $\rho_{\rm DM}(\bm{x})$. The initial step is the reconstruction of the distribution of DM in real space, $\rho_{\rm DM}$, using the sparse halo number density field in redshift space, $\rho_{s}$. Each channel of the input contains halos within a specific mass range. The DM halos (and subhalos) are catalogued in descending order of mass and divided into four mass intervals, with boundaries defined by the logarithmic mass ratio as follows: $\log_{10}(M/M_{\odot})\in[15.01,13.30,12.56,12.31,12.17]$, corresponding to four number densities of halos $\rho_{h}\in[0.0001,0.001,0.002,0.003]~{}(h/{\rm Mpc})^{3}$. Note that the mass information may be estimated to a reasonable degree of accuracy using empirical formulas, which allow for the inference of the masses of these objects from the observed apparent magnitudes of galaxies. The target data is the three-dimensional number density field of DM in real space. Our UNet model is constructed for the purpose of reconstructing the real-space $\rho_{\rm DM}$.

Two mass-weighting schemes are adopted for comparison: i) “with $M_{\rm halo}$ weighting”: in this scheme, DM halos are divided into the four intervals based on mass, and each interval is assigned a weight proportional to its mass.

ii) “without $M_{\rm halo}$ weighting”: in this scheme, DM halos are also divided into the four intervals based on mass, but no weights are assigned to these intervals.

The first scheme assumes an ideal scenario where we have complete knowledge of the mass information of DM halos. In contrast, the second one better reflects current observational capabilities, allowing only rough estimates of the mass distribution of galaxies, and thus corresponds more closely with actual observations. To assess the impact of mass information accuracy, results from both schemes will be presented simultaneously.

2) The mapping from a combination of $\rho_{s}(\bm{x})$ and $\rho_{\rm DM}(\bm{x})$ to $\bm{v}(\bm{x})$ or $\bm{m}(\bm{x})$ is achieved using two UNet models. These models are constructed to reconstruct the direction and magnitude of the DM velocity and momentum fields in real space. The UNet models utilize $\rho_{s}$, $\rho_{\rm DM}$, and the linear velocity in redshift space, $\bm{v}_{\rm lin}$, derived from linear perturbation prediction as inputs.

In light of the neural network model proposed by (Wu et al., 2021), we employed the UNet architecture for the construction of our model. The architecture of our neural network and its components are depicted in Fig. 2. The input is data blocks with an arbitrary number of channels, with each channel corresponding to a specific mass range of $\rho_{s}$.

Given that the velocity field is decomposed into the velocity magnitude and direction, we constructed two structurally similar neural networks to handle them separately. The networks conclude with an output layer comprising 1+3 channels. The three channels correspond to the components of the velocity direction, while the fourth channel corresponds to the velocity magnitude. The final step in the reconstruction process is the combination of the results from all output channels, which ultimately yields the complete reconstruction of the three-dimensional velocity field.

The notations in Fig. 2, “${n_{\rm in}}$” and “${n_{\rm out}}$”, represent the number of input and output channels, respectively. These values may vary across different reconstruction tasks, as shown in Tab. 1.

The color blocks represent the various operations performed by the neural network, with arrow lines connecting the input and output. The dimensions of the input, intermediate, and output fields (channels $\times$ spatial pixels) are defined, along with the size and number of labeled 3D convolutional kernels (“conv”).

Given the constraints of GPU memory, training time, and model size, we divided a large box with a physical scale of 1200 ${\rm Mpc}/h$ into $4^{3}$ small boxes with a physical scale of 300 ${\rm Mpc}/h$, each containing $128^{3}$ grid points in the CIC interpolation scheme. A total of 32 small boxes were utilized for training and 32 for validation purposes. The neural network model receives inputs and generates outputs targeted at these small boxes, which consist of $128^{3}$ pixels each.

During the testing phase, We applied the neural network to large boxes, each consisting of $256^{3}$ pixels, as the input data for the model. The output also contains $256^{3}$ pixels, corresponding to a physical size of 600 ${\rm Mpc}/h$. Using the neural network on large boxes helps us eliminate less accurate outputs at the edges, which can be a significant issue when applying the UNet model to smaller boxes. For training, the $k_{\rm min}$ for small boxes is 0.026, while the $k_{\rm min}$ for validation with large boxes is 0.013.

Some points require further clarification, which will be detailed below.

1. 1)

   To modify the field size, the $\emph{stride}=2$ parameter can be employed, which allows for a reduction or increase in the field size by a factor of two.

2. 2)

   The “init” 3D convolutional layer enables the network to rapidly learn large-scale information due to its sufficiently large receptive field.

3. 3)

   The convolutional layer, which is situated at the output of the network, serves to increase the network’s complexity. In order to prevent overfitting, a dropout layer is incorporated at the end of the convolutional layer. This layer also alters the number of channels.

4. 4)

   The incorporation of batch normalization (BN) layers into the neural network facilitates the acceleration of training convergence and the prevention of overfitting. Additionally, the implementation of rectified linear units (ReLU) as activation layers subsequent to convolutional layers enhances the network’s nonlinearity.

5. 5)

   The trained UNet enables the prediction of $\rho_{\rm DM}$ by inputting $\rho_{s}$. This, in turn, enables the prediction of the real-space fields, including the velocity ($\bm{v}$) and momentum ($\bm{m}$) fields, and the associated clustering statistics, which can be directly measured.

6. 6)

   Additionally, due to the constraints of the limited dimensions of the training box, the model struggles to learn large-scale velocity modes. To mitigate potential bias from the small box dataset, the velocity field predicted by traditional linear reconstruction is used as one of the UNet inputs to reconstruct the velocity or momentum field. The linear velocity and momentum predictions in redshift space are determined through

   $$\bm{v}_{\rm lin}(\bm{k})=afH\frac{i\bm{k}}{k^{2}}\frac{\delta_{s}(\bm{k})}{\beta}\,,$$

   (2)

   and

   $$\bm{m}_{\rm lin}=(1+\frac{\delta_{s}}{\beta})\bm{v}_{\rm lin}\,.$$

   (3)

   Here $\delta_{s}$ is divided by the linear bias $\beta$, where $\beta=$ 1.85 for “without $M_{\rm halo}$ weighting” scheme and 2.57 for “with $M_{\rm halo}$ weighting” scheme and is determined by the ratio of the integrated DM power spectrum to the integrated halo power spectrum over the relevant $k$-range of interest, i.e.,

   $$\beta^{2}=\frac{\int_{k<0.3}\frac{P_{\rm halo}(k)}{P_{\rm DM}(k)}dk}{\int_{k<0.3}dk}\,.$$

   (4)

7. 7)

   The Tree Parzen Estimator (TPE) process (Bergstra et al., 2011), based on Bayesian optimization, is employed to optimize hyperparameters, with the Python package Optuna¹¹1https://optuna.readthedocs.io utilized to maximize the model accuracy (Akiba et al., 2019). The hyperparameters that are optimized include: 1) the learning rate; 2) dropout percentage; 3) the number of training data sets; 4) the number of channels in the intermediate layer, and 5) the training batch size. The hyperparameters are tuned by searching for values that minimize the validation loss of the model. A minimum of five experiments were conducted for each set of hyperparameter values, with the objective of evaluating the impact of these values on the performance.

The objective of the training for UNet is to minimize the loss between the prediction for a given field and the simulation truth for each voxel.

1. 1)

   For training our UNet model to reconstruct the real-space DM density fields $\rho_{\rm DM}$, we use the Mean Squared Error (MSE) loss function,

   $$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}\big{(}\rho_{i}-\rho^{\rm true}_{i}\big{)}^{2}\,,$$

   (5)

   where the index $i$ runs over all $N$ pixels of a field.

2. 2)

   To account for the contributions of the velocity magnitude ($v\equiv|\bm{v}|$) and the velocity direction (unit vector $\hat{v}\equiv\bm{v}/|\bm{v}|$), we choose the following two-term loss function,

   $$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}\left[\frac{1}{4}(|v_{i}|-|v_{i}^{\rm true}|)^{2}+\frac{3}{4}\left(1-\cos\phi_{i}\right)\right]\,,$$

   (6)

   where $\cos\phi_{i}\equiv\hat{v}_{i}\cdot\hat{v}_{i}^{\rm true}$, and the index $i$ denotes the $i$-th pixel. As observed, the first term is responsible for $|\bm{v}|$ and corresponds to the MSE loss, which is essentially equivalent to the maximum likelihood solution under the Gaussian constant variance assumption. The second term, of course, measures the deviation between the reconstructed and true values of $\hat{v}$. The coefficients of these two terms can be considered as normalization factors and are determined by the number of channels: 1 for the magnitude $|\bm{v}|$ and 3 for the three directions, i.e., $\hat{v}_{x}$, $\hat{v}_{y}$, and $\hat{v}_{z}$. Empirically, this loss function has proven to be stable and effective during our training process, yielding good results in velocity (momentum) reconstruction.

   

3. 3)

   We trained our UNet using the popular Adam algorithm (Kingma & Ba, 2014) for training deep neural networks. This algorithm iteratively reduces the training loss by calculating its gradient with respect to the model parameters and taking a small step in the direction of maximum reduction. From Fig. 3, it can be seen that both the velocity model and the momentum model converge after approximately 1000 epochs of training.

The main objective of this study is to compare the UNet-predicted DM velocity/momentum in real space with the ground truth. In order to facilitate the analysis of the data, it is first necessary to describe the statistics that will be employed throughout the study. The two-point correlation function is a commonly used statistical measure for characterizing a density field,

where the density contrast field, denoted by $\delta(\bm{x})$, is a function of any point $\bm{x}$. The separation vector, $\bm{r}$, is a vector that represents the distance between two points. The ensemble mean, represented by $\langle\cdot\rangle$, is computed by averaging over all points $\bm{x}$ in a spatial mean. The power spectrum of $\delta(\bm{x})$ is related to the Fourier transform of the correlation function, $\xi(\bm{r})$, by a simple mathematical identity:

where the three-dimensional wavevector of the plane wave, denoted by $\bm{k}$, has magnitude $k\equiv|\bm{k}|$, related to the wavelength $\lambda$ by $k=2\pi/\lambda$.

As with the scalar field $\delta$, we can also define power spectra for various velocity fields of interest. The velocity field, $\bm{v}$, is completely described by its divergence, $\theta\equiv\nabla\cdot\bm{v}$, and its vorticity, $\bm{\omega}=\nabla\times\bm{v}$, via the Helmholtz decomposition. In Fourier space, they become purely radial and transversal velocity modes, respectively, defined by $\theta(\bm{k})=i\bm{k}\cdot\bm{v}(\bm{k})$ and $\bm{\omega}(\bm{k})=i\bm{k}\times\bm{v}(\bm{k})$. The power spectra of the velocity, divergence, vorticity, and velocity magnitude are given by

where indices $i$ and $j$ denote the components in the Fourier space coordinates.

According to the linear perturbation theory, the continuity equation leads to the following relationship: $\theta=-\mathcal{H}f\delta$, where $\mathcal{H}=aH$ is the conformal Hubble parameter, $a$ represents the cosmic scale factor, $f$ is the linear growth rate, defined as $f=d\ln D/d\ln a$, with $D$ being the linear density growth factor. Under the $\Lambda$CDM model, the growth rate is approximately given by $f\approx\Omega_{m}^{0.55}$ (Linder, 2005).

Fig. 4 presents a comparison between the velocity magnitude and direction reconstructed by UNet and the ground truth values, using the “without $M_{\rm halo}$ weighting” weighting scheme. As observed, the reconstructed velocity magnitude and the true velocity magnitude are $297.22\pm 135.31$ and $294.60\pm 131.02$ km/s, respectively. Additionally, the reconstructed momentum magnitude is $315.17\pm 209.29$ km/s, while the true one is $309.50\pm 213.43$ km/s. The two mean values are consistent with the true values across all slices, with significant small deviations. When considered their statistical uncertainties, these deviations are negligible.

Moreover, by defining the quantity as $\cos\phi_{i}\equiv\hat{v}_{i}\cdot\hat{v}^{\rm true}_{i}$, where $\hat{v}_{i}$ is the velocity direction reconstructed by the model and $\hat{v}^{\rm true}_{i}$ is the true velocity direction, one can measure the deviation between the angles. The results of the tests conducted among these slices indicate that $1-\cos\phi_{i}$ is $0.01\pm 0.06$ for velocity and $0.04\pm 0.11$ for momentum, respectively. This implies that the directions of the reconstructed velocity fields are nearly identical to those of the true velocity field, and that our model performs high reconstruction accuracy.

Furthermore, Fig. 5 illustrates a comparison between the reconstructed velocity/momentum divergence and the true velocity/momentum divergence. The divergence of velocity reconstructed by UNet and the true value are $0.40\pm 18.49$ and $0.92\pm 17.55$ km/s, respectively. The divergence of momentum reconstructed by UNet and the true one are $0.34\pm 25.83$ and $0.34\pm 25.79$ km/s, respectively. At the same time, each of the histogram demonstrates that the reconstructed probability distribution is similar to that of the truth. The findings suggest that the discrepancies in the reconstruction for velocity/momentum divergence fields are well consistent with the true values, with an accuracy exceeding 1% relative to the statistical uncertainty.

Furthermore, let us introduce the metrics that will be employed for evaluating the reconstruction accuracy. For an arbitrary reconstructed field of halos from a UNet, denoted by the shorthand notation $f$, where $f\in\{\bm{v},\theta_{v}\}$ for velocity and its divergence, and $f\in\{\bm{m},\theta_{m}\}$ for momentum and its divergence, the following metrics are employed to describe the relative deviation and correlation coefficient, respectively, in order to compare the reconstructed field with the true one.

The relative deviation, $R$, and the correlation coefficient, $C_{r}$, are defined as follows:

where $\mathcal{O}_{f}$ represents an arbitrary observable for $f$. $C_{r}$ is defined between the reconstructed field $f$ and the true field $f^{\prime}$, both of which have the same total number of pixels, $N_{\rm pix}$. The sample mean and standard deviation of field $f$ are denoted by $\bar{f}$ and $\sigma_{f}$, respectively. It is evident that both metrics provide a physical insight for comparison, such that the ideal reconstruction is equivalent to $R=0$ and to $C_{r}=1$.

The coefficients $C_{r}$ and relative deviation $R$ for various velocity components are calculated using Eq. 10. The calculations are based on averaging over four test sets, each with a box size of 300 ${\rm Mpc}/h$ on each side. The results are summarized in Tab. LABEL:tab:rc. It can be observed that in all cases, the $C_{r}$ values for all fields, except the momentum divergence field, are at the level of 0.9. Additionally, the changes induced by the mass-weighting scheme on $C_{r}$ are not significant. The velocity reconstruction appears to exhibit slightly superior performance compared to the momentum reconstruction. Moreover, the $R$ values approach zero, indicating that the reconstructed field has minimal bias compared to the true one. When comparing the ”with $M_{\rm halo}$ weighting” scheme to the “without $M_{\rm halo}$ weighting” scheme, we find that even without complete knowledge of the halo mass, we can still achieve accurate reconstruction of the field.

In order to provide a more comprehensive visual representation of the reconstruction performance, Fig. 6 depicts the joint probability distributions of the reconstructed fields and the true fields for magnitudes of velocity and momentum. This illustration compares the results obtained with the two mass weighting schemes. For each of the fields, the distributions were calculated using a large box with a side length of $600~{}{\rm Mpc}/h$. As can be observed, the predicted distributions for each field exhibit a strong correlation with the true one, falling very close to the diagonal. These findings suggest that our neural network performs an effective reconstruction. Furthermore, when employing the “without $M_{\rm halo}$ weighting” weighting scheme, even without knowledge of the specific halo mass values, our UNet model is still capable of accurately reconstructing the fields, closely matching the true values. This indicates that the network has learned the information about the mapping from the sparse halo number density field in redshift space to the velocity/momentum fields in real space. As demonstrated in Fig. 7 for their divergence fields, similar findings are evident.

To further test the difference between the reconstructed velocity/momentum fields and the true ones, the joint probability distribution of density-divergence and density-velocity (momentum) are shown in Fig. 8 for the weighting scheme of “with $M_{\rm halo}$ weighting” and Fig. 9 for “without halo mass”. It is evident that the predicted results are in strong agreement with the ground truth across a wide range of $\delta_{\rm DM}$ values, even in high-density regions where $\delta_{\rm DM}\gtrsim 4$. These high-density areas are sparse and highly non-linear, yet the model still performs effectively.

As illustrated in Fig. 10, the resulting DM density power spectra in real space, calculated using Eq. 8 for various cases, are summarized. Two mass weighting schemes of “with $M_{\rm halo}$ weighting” and “without $M_{\rm halo}$ weighting” are presented in the left and right panels, respectively, for comparison. The auto power spectra for the predicted density (blue) and the true density (black) are shown, along with the auto power spectrum of sparse DM halo (red), providing a detailed comparison.

To accurately estimate the statistical uncertainty $\Delta P(k)$, the number of independent Fourier modes is taken into account, which leads to a rescaling of the standard deviation $\delta P(k)$ (estimated over the 25 test boxes) via

Here, $V_{\rm all}$ represents the total independent volume, while $V_{\rm overlap}$ denotes the overlap volume for the calculated power spectrum. Since the boxe have a physical size of $1200\times 1200\times 600~{}({\rm Mpc}/h)^{3}$, divided into 25 test boxes with a size of 600 ${\rm Mpc}/h$ on each side, the value of $\sqrt{V_{\rm all}/V_{\rm overlap}}$ is 0.4.

The relative deviation $R$ as defined in Eq. 10 are illustrated in the bottom panels. For “without $M_{\rm halo}$ weighting” case, the discrepancies between the measured auto spectra and the true auto spectrum are significant small, essentially invisible in the left panel when $k<0.3~{}h/{\rm Mpc}$. It can be observed that the discrepancy across this scale range is $|R|<0.13$, indicating a good recovery of both the amplitude information of the field, consistent with the findings in (Wang et al., 2024).

In comparison to the left panels, the reconstructed power spectra for “with $M_{\rm halo}$ weighting” exhibit relatively larger deviations. The maximum deviation is $R=0.15$ for auto correlation. By comparing these results, we demonstrate that the absence of this information does not significantly reduce reconstruction accuracy.

Furthermore, we now examine the power spectrum reconstruction for velocity, momentum, and their divergences. In Fig. 11, the magnitudes of the momentum and velocity power spectra are presented. The auto correlation power spectra between the predicted and true fields based on the two weighting schemes are also shown for comparison. The estimated curves and shaded regions are based on the mean and standard deviation of the derived power spectra from a total of 25 boxes in the test dataset, each with a side length of $600~{}{\rm Mpc}/h$.

It can be observed that the auto correlation power spectra of the velocity and momentum can be reconstructed with a relative deviation of $|R(k)|<0.07$ for $k\in[0.05,0.2]~{}h{\rm Mpc}^{-1}$. In contrast, the velocity and momentum spectra have $|R(k)|\in[0,0.13]$, indicating an overestimate in the scales of $k\in[0.03,0.05]~{}h{\rm Mpc}^{-1}$. In general, the reconstruction performance for “with $M_{\rm halo}$ weighting” across all scales appears better than that for “without $M_{\rm halo}$ weighting”. This is evidenced by the fact that even without complete knowledge of the halo mass, we can still accurately reconstruct the auto-correlation of the DM velocity field and momentum field using UNet.

Moreover, since the momentum field is a density-weighted velocity, it is expected to be more sensitive to the reconstruction accuracy of the DM density field. However, the reconstruction accuracy for the momentum is comparable to that of the velocity, thanks to the good reconstruction of the DM density, as illustrated in Fig. 10. It is clear that as the reconstruction uncertainty from the DM density field increases, the statistical error in the momentum and its divergence power spectra will also increase.

Moreover, as shown in Fig. 12, the auto power spectra of the velocity and momentum divergence yield $R(k)<0.06$ for scales of $k$ from $0.05$ to 0.1 $h/{\rm Mpc}$. At $k>0.05$ $h/{\rm Mpc}$, the auto-correlation of velocity divergence agrees well with the true power spectrum within $2\sigma$, while the auto-correlation of momentum divergence slightly overestimates the true value. This discrepancy may arise from the inaccurate reconstruction of the DM density field, likely due to the limited number of large-mass halos in the current training samples.

In addition, a comparison of the results obtained by the two different weighting schemes reveals that there is not a significant difference between them. This suggests that precise mass information is not necessary for the velocity/momentum reconstruction. Overall, our UNet model demonstrates satisfactory performance on linear and nonlinear scales of $k\in[0.03,0.3]~{}h/{\rm Mpc}$. These results highlight the ability of the UNet model to learn various velocity quantities from the halo number density field on the nonlinear scales, which is an important finding in this study. In light of the predicted power spectra, it can be reasonably concluded that the UNet model’s predictions are in relatively good agreement with the truth.

To further validate the effectiveness of the UNet model, we will demonstrate comparisons between the reconstructed measurements and the simulation truth in real space. As the input observables in the neural network are in redshift space, the relevant comparison can be used to assess the accuracy of the peculiar velocities corrected. The following analysis was conducted based on a total of 18 simulation boxes, each with a side length of $600~{}{\rm Mpc}/h$.

In Fig. 13, a comparison is performed between the two-dimensional anisotropic 2PCFs, $\xi(r_{\perp},r_{\|})$, of density fields between the prediction and the simulation truth in real space. As illustrated by the red dotted contours, the 2PCF measured in redshift space without any RSD corrections demonstrates the significant impact of RSD effects. The Kaiser effect results in the “squashed” structure along LoS, while the random velocities on the non-linear small scales produce the so-called “fingers-of-God” (FoG) effect, which causes structures to be elongated along the LoS. The velocity field reconstructed by UNet was utilized to correct the velocities of the halo. Note that the corrected 2PCF exhibits a high degree of agreement with the real-space 2PCF across all scales. Furthermore, the 2PCF corrections for both Kaiser and FoG effects are achieved with a high degree of accuracy. The successful corrections for the FoG effect indicate that the UNet is capable of accurately predicting the DM field in real space.

In order to accurately assess the statistical properties of an arbitrary real-space field, it is necessary to consider the higher-order power spectrum multipoles. In general, the power spectrum in redshift space $P(k,\mu)$, where $\mu$ is the cosine of the angle between the wavevector $\bm{k}$ and the LoS direction, can be expressed by expanding its multipole components:

where $\mathcal{L}_{\ell}$ are the Legendre polynomials of order $\ell$. In this study, we focus on the Legendre polynomials of order $\ell$, specifically the quadrupole ($\ell=2$) and the hexadecapole ($\ell=4$), denoted as $P_{2}$ and $P_{4}$, respectively.

In Fig. 14, we compare the power spectrum multipoles of the UNet-reconstructed DM density field with the true values. The top panels depict the measured $P_{2}$ in redshift space (red), the UNet-reconstructed DM field (blue) in real space, and the real-space true value (black). Two mass weighting schemes were employed for the purpose of demonstrating the reconstruction performance: “without $M_{\rm halo}$ weighting” (left) and “with $M_{\rm halo}$ weighting” (right). The shaded regions represent the $1\sigma$ confidence interval, estimated based on the test dataset.

As demonstrated, the RSD effects can markedly contribute to anisotropic clustering, resulting in an amplitude of $P_{2}(k)$ that is approximately one to two orders of magnitude greater than that observed in real space across all $k$ values of interest. The quadrupole of the UNet-predicted DM density field in real space can significantly suppress the anisotropy clustering, yielding a reconstructed result that is in good agreement with the true one at the $2\sigma$ level when $k\lesssim 0.4~{}h/{\rm Mpc}$. Furthermore, it has been found that the “with $M_{\rm halo}$ weighting” scheme yields a slightly more accurate reconstruction. The discrepancy between the reconstructed and true values averaged over the $k$ range is $174.29\pm 702.27$ for the “with $M_{\rm halo}$ weighting” scheme and $175.05\pm 711.88$ for the “without $M_{\rm halo}$ weighting” scheme. This indicates that we can reconstruct the quadrupole of the DM density field without complete knowledge of halo mass information.

The lower panels of Fig. 14 depict a similar comparison of a higher-order multipole, the hexadecapole, $P_{4}$. As a consequence of its higher order, the statistical uncertainty for halos in redshift space appears to exceed that of $P_{2}$. Nevertheless, the UNet is still capable of successfully reconstructing $P_{4}$ of the DM density field in real space, with the associated statistical uncertainty being approximately comparable to that of $P_{2}$. The discrepancy between the reconstructed and true values falls within the $2\sigma$ level. Furthermore, the use of a weighting scheme based on halo mass can result in a slight improvement in reconstruction accuracy, with the average deviation of $-60.03\pm 543.83$ reduced by $27\%$ compared to the “without $M_{\rm halo}$ weighting” case.

The proposed UNet is capable of automatically applying RSD corrections. Furthermore, the reconstructed multipoles of both the quadrupole and hexadecapole have been found to agree with the true values at the $2\sigma$ level, within the range of $k\in[0.06,0.3]~{}h/{\rm Mpc}$. Moreover, the “with $M_{\rm halo}$ weighting” scheme has been observed to yield higher reconstruction accuracy and smaller statistical uncertainty.

In conclusion, while the RSD effects can significantly contribute to anisotropic clustering, distorting $P_{2}(k)$ and $P_{4}(k)$, our proposed UNet effectively corrects for these RSD effects and yields accurate predictions for the real-space statistical measurements of density. The reconstructed results align well with the true values at the $2\sigma$ level for $k\in[0.03,0.4]~{}h/{\rm Mpc}$. Furthermore, the “with $M_{\rm halo}$ weighting” scheme provides additional information, resulting in a slightly more accurate reconstruction.