hypersurface defined by the _tropical polynomial_

$\displaystyle\varphi\colon M^{\vee}_{\mathbb{R}}\to\mathbb{R},\
\varphi(m)=\max\\{\langle m,n\rangle-\mu(n)\ |\ n\in\Delta^{\vee}\\}.$

It is known that $\Pi_{\Sigma}$ is a deformation retract of $\Pi_{t}$.
According to [Mik04, Corollary 6.4], when $t\to\infty$ the rescaled amoeba
$\Pi_{t}/\log t$ converges to $\Pi_{\Sigma}$. Combinatorially, $\Pi_{\Sigma}$
is the dual cell complex of $\mathcal{T}$. In particular, the set of connected
components of $\mathbb{R}^{n+1}\setminus\Pi_{\Sigma}$ bijectively corresponds
to the set $\operatorname{Vert}(\mathcal{T})$ of vertices of $\mathcal{T}$
according to which $\alpha\in\operatorname{Vert}(\mathcal{T})$ achieves the
maximum of $\langle m,\alpha\rangle-\mu(\alpha)$ for
$m\in\mathbb{R}^{n+1}\setminus\Pi_{\Sigma}$. Note that
$\mathbb{R}^{n+1}\setminus\Pi_{t}$ for $t\gg 0$ has the same combinatrics as
$\mathbb{R}^{n+1}\setminus\Pi_{\Sigma}$. In the sequel, we will fix a general
$t\gg 0$ and drop $t$ from the notation.

Let $\operatorname{ret}\colon\Pi\to\Pi_{\Sigma}$ be the continuous map induced
by the retraction. Then the composition

(6.3) $\displaystyle
H\xrightarrow{\operatorname{Log}|_{H}}\Pi\xrightarrow{\operatorname{ret}}\Pi_{\Sigma}$

gives the $A$-side SYZ fibration. Recall that $H$ admits a pants decomposition
[Mik04, Theorem 1’]. By [Mik04, Proposition 4.6] the $k$-th intersection of
$i_{1},\cdots,i_{k}$-th legs [GS2, Definition 5.2.2] of an $n$-dimensional
tailored pants $\tilde{P}_{n}$ is isomorphic to a product

$\displaystyle\mathbb{C}^{*}_{z_{i_{1}}}\times\cdots\times\mathbb{C}^{*}_{z_{i_{k}}}\times\tilde{P}_{n-k}.$

Under (6.3) a $k$-th intersection of legs maps to a subset of $k$-stratum away
from lower dimensional strata. Here, we equip $\Pi_{\Sigma}$ the canonical
stratification. In particular, the fiber over a general point of a $k$-stratum
contains $T^{k}$. On the other hand, the $B$-side SYZ fibration is induced by
the map from the total space of the anticanonical sheaf on
$\mathbf{T}_{\Sigma}=X_{\Sigma}$ defined as (6.1).

Now, we return to the case where $\Sigma$ is a smooth quasiprojective stacky
fan. Since $\mathcal{T}$ is star-shaped, each $(n+1)$-simplex is a polytope

$\displaystyle\Delta^{\vee}_{\sigma}=\operatorname{Conv}(0,\alpha^{1},\ldots,\alpha^{n})\subset
M_{\mathbb{R}}$

spanned by the origin and primitive ray generators
$\alpha^{1},\ldots,\alpha^{n}$ of some maximal dimensional cone $\sigma$.
Consider the map

$\displaystyle\Delta^{\vee}_{n+1}\to\Delta^{\vee}_{\sigma},\
e_{i}\mapsto\alpha^{i}$

from the standard $(n+1)$-simplex with the standard basis
$e_{1},\ldots,e_{n}$, whose dual induces a map
$f_{\Delta^{\vee}_{\sigma}}\colon\mathbb{T}^{\vee}_{\mathbb{C}}\to\mathbb{T}^{\vee}_{\mathbb{C}}$.

######  Definition 6.12 ([GS2, Definition 5.1.4]).

The _$\Delta^{\vee}_{\sigma}$ -pants_ is the inverse image
$\tilde{P}_{\Delta^{\vee}_{\sigma}}=f^{-1}_{\Delta^{\vee}_{\sigma}}(\tilde{P}_{n})$
of the tailored pants. We write $\tilde{A}_{\Delta^{\vee}_{\sigma}}$ for its
amoeba $\operatorname{Log}(\tilde{P}_{\Delta^{\vee}_{\sigma}})$. For
$b=(b_{1},\ldots,b_{n+1}),b_{i}\gg 0$ we denote by
$\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}}$ and
$\tilde{A}^{b}_{\Delta^{\vee}_{\sigma}}$ respectively the _translated tailored
$\Delta^{\vee}_{\sigma}$-pants_ obtained by scaling the coefficients of the
defining polynomial by $e^{-b_{i}}$ and its amoeba obtained by translation to
the first orthant.

######  Lemma 6.13 ([GS2, Lemma 5.3.6]).

Let $\partial^{0}\tilde{A}^{b}_{\Delta^{\vee}_{\sigma}}$ be the component of
$\partial\tilde{A}^{b}_{\Delta^{\vee}_{\sigma}}$ which bounds the region of
$\mathbb{R}^{n+1}$ containing the inverse image of the all-negative orthant.
Restrict the canonical Liouille structure on $T^{*}\mathbb{T}^{\vee}$ to
$\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}}$. Let $\Sigma_{\sigma}\subset\Sigma$
be the stacky subfan whose primitive ray generators are
$\alpha_{1},\ldots,\alpha_{n}$. Then we have

$\displaystyle\operatorname{Core}(\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}})=\operatorname{Log}^{-1}(\partial^{0}\tilde{A}^{b}_{\Delta^{\vee}_{\sigma}})\cap\mathbb{L}(-\Sigma_{\sigma}).$

From [GS2, Theorem 6.2.4] it follows that $\operatorname{Core}(H)$ is the
gluing of $\operatorname{Core}(\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}})$ for
all $\sigma\in\Sigma_{\max}$. Hence in this case the fiber of (6.3) over a
point of $k$-stratum away from lower dimensional strata becomes isomorphic to
a finite cover of a product

$\displaystyle\mathbb{C}^{*}_{z_{i_{1}}}\times\cdots\times\mathbb{C}^{*}_{z_{i_{k}}}\times\tilde{P}^{b}_{\Delta^{\vee}_{\sigma/\langle
i_{1},\ldots,i_{k}\rangle}}.$

Here, $\tilde{P}^{b}_{\Delta^{\vee}_{\sigma/\langle
i_{1},\ldots,i_{k}\rangle}}$ is the $k$-th intersection of
$i_{1},\cdots,i_{k}$-th legs [GS2, Definition 5.2.5] of
$\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}}$ with

$\displaystyle\Delta^{\vee}_{\sigma/\langle
i_{1},\ldots,i_{k}\rangle}=\operatorname{Conv}(0,\alpha^{1},\ldots,\hat{\alpha}^{i_{1}},\ldots,\hat{\alpha}^{i_{k}},\ldots,\alpha^{n}).$

On the other hand, the $B$-side SYZ fibration becomes the composition of the
structure morphism $\mathbf{T}_{\Sigma}\to X_{\Sigma}$ to the coarse moduli
space with the map defined as (6.1). The fiber over any point of a $k$-stratum
is the real part of the corresponding subgroup of the Deligne–Mumford torus
[FMN10, Definition 2.4, Proposition 2.6] acting on $\mathbf{T}_{\Sigma}$.

###  6.4. Proof of Theorem 1.4

For a smooth quasiprojective stacky fan $\Sigma\subset M_{\mathbb{R}}$,
consider the fanifold $\Phi=\Sigma\cap S^{n+1}$ from [GS1, Example 4.22]
generalized as in [GS1, Section 6]. In particular, to each $0$-stratum $P$ we
associate the stacky fan $\Sigma_{P}=\Sigma/\rho_{P}$ where
$\rho_{P}\in\Sigma$ is the ray passing through $P$. Hence $\mathbf{T}(\Phi)$
coincide with the toric boundary divisor $\partial\mathbf{T}_{\Sigma}$ of
$\mathbf{T}_{\Sigma}$. Fix a stratified homeomorphism
$h_{\Phi}\colon\partial\Delta^{\vee}\to\Phi$. Then strata of
$\partial\Delta^{\vee}$ inherit the labels from $\Phi$.

######  Definition 6.14.

Let $\Pi^{0}_{\Sigma}$ be the connected component of
$\mathbb{R}^{n+1}\setminus\Pi_{\Sigma}$ corresponding to the origin of
$\Delta^{\vee}$. We define $\Psi$ as the image of a fixed stratified
homeomorphism $h_{\Psi}\colon\partial\Pi^{0}_{\Sigma}\hookrightarrow
S^{n+1}\subset M^{\vee}_{\mathbb{R}}$, which makes the inherited labels on
strata of $\Psi$ compatible with that on strata of $\Phi$ transported to
$M^{\vee}_{\mathbb{R}}$ via a fixed inner product.

Note that $\partial\Pi^{0}_{\Sigma}$ is the image of the composition

$\displaystyle\partial\mathbf{T}_{\Sigma}\hookrightarrow\mathbf{T}_{\Sigma}\to
X_{\Sigma}\to(X_{\Sigma})_{\geq 0}\to M^{\vee}_{\mathbb{R}}.$

Hence $\Phi$ satisfies (vi) and admits the dual stratified space $\Psi$. In
particular, the $B$-side fibration $\mathbf{T}(\Phi)\to\Psi$ is the gluing of
the compositions $\mathbf{T}_{\Sigma_{P}}\to X_{\Sigma_{P}}\to Q_{P}$ and
compatible with that $\partial\mathbf{T}_{\Sigma}\to\partial\Pi^{0}_{\Sigma}$
for very affine hypersurface. Namely, we have the commutative diagram

$\displaystyle\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
14.11113pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbf{T}(\Phi)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern
0.0pt\raise 1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 0.0pt\raise
1.0pt\hbox{\lx@xy@droprule}}\hbox{\kern
0.0pt\raise-1.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.44223pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
39.85pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\partial\mathbf{T}_{\Sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
51.13786pt\raise-28.63779pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Psi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.88571pt\raise-33.51721pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{h^{-1}_{\Psi}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
38.11113pt\raise-40.27554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.11113pt\raise-40.27554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\partial\Pi^{0}_{\Sigma}.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

######  Lemma 6.15.

There is a stratified homeomorphism of the image of $\operatorname{Core}(H)$
under the composition $\operatorname{ret}\circ\operatorname{Log}$ and $\Phi$
transported to $M^{\vee}_{\mathbb{R}}$ via the fixed inner product.

###### Proof.

As explained above, $\operatorname{Core}(H)$ is the gluing of
$\operatorname{Core}(\tilde{P}^{b}_{\Delta^{\vee}_{\sigma}})$ for all
$\sigma\in\Sigma_{\max}$. By Lemma 6.13 each piece is the Legendrian boundary
$\partial_{\infty}\mathbb{L}(-\Sigma_{\sigma})$ of the FLTZ Lagrangian
$\mathbb{L}(-\Sigma_{\sigma})$ associated with the stacky subfan
$\Sigma_{\sigma}\subset\Sigma$. Unwinding the proof of [Nad, Theorem 5.13],
one sees that $\operatorname{ret}\circ\operatorname{Log}$ projects each
$k$-dimensional torus in $\partial_{\infty}\mathbb{L}(-\Sigma_{\sigma})$ and
the zero set of the Liouville vector field to a point of the corresponding
cone in $\Sigma_{\sigma}(n+2-k)$ transported to $M^{\vee}_{\mathbb{R}}$. Such
$k$-dimensional tori are connected with lower dimensional tori in
$\partial_{\infty}\mathbb{L}(-\Sigma_{\sigma})$ and the zero set of the
Liouville vector field by the Liouville flow. Hence the composition

$\displaystyle
h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}|_{\operatorname{Core}(H)}\colon\operatorname{Core}(H)\to\Psi$

projects the $k$-dimensional torus to a point of the corresponding
$(n+1-k)$-stratum of $\Phi$ transported to $M^{\vee}_{\mathbb{R}}$. Note that
tori in different $\partial_{\infty}\mathbb{L}(-\Sigma_{\sigma})$ mapping to
the same point get identified in $\operatorname{Core}(H)$. The images of
$k$-dimensional tori are connected with that of lower dimensional tori by the
images of the Liouville flow, which become parallel to the corresponding
strata of $\Phi$. Thus one can find a homeomorphism of the image of each
stratum of $\operatorname{Core}(H)$ under
$h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}$ to the corresponding
stratum of $\Phi$ transported to $M^{\vee}_{\mathbb{R}}$. By construction such
homomorphisms glue to yield the desired map. ∎

######  Corollary 6.16.

There is a diffeomorphism of $\widetilde{\mathbb{L}}(\Phi)$ and
$\operatorname{Core}(H)$ over $\Phi$ transported to $M^{\vee}_{\mathbb{R}}$.

###### Proof.

For any $k$-stratum $S^{(k)}_{P}$ of $\Phi$ adjacent to a $0$-stratum $P$, its
inverse image under
$h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}|_{\operatorname{Core}(H)}$
is diffeomorphic to the zero section of $T^{*}\widehat{M}_{S^{(k)}_{P}}\times
T^{*}S^{(k)}_{P}$. By construction

$\displaystyle(h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}|_{\operatorname{Core}(H)})^{-1}(S^{(k)}_{P_{\circ}}),\
\widehat{M}_{S^{(k)}_{P}}\times S^{(k)}_{P,\circ}$

respectively glue along the boundaries to yield
$\operatorname{Core}(H),\widetilde{\mathbb{L}}(\Phi)$. ∎

######  Corollary 6.17.

There is a symplectomorphism of pairs

(6.4)
$\displaystyle(\widetilde{\mathbf{W}}(\Phi),\widetilde{\mathbb{L}}(\Phi))\hookrightarrow(H,\operatorname{Core}(H))$

over $\Psi$ with respect to $\underline{\pi}$ and the modification of
$h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}$ by further retraction
$\Pi_{\Sigma}=\operatorname{ret}\circ\operatorname{Log}(H)$ to
$\partial\Pi^{0}_{\Sigma}$ before composing $h_{\Psi}$.

###### Proof.

Cotangent scaling makes $(T^{*}\widehat{M}_{S^{(k)}_{P}}\times
T^{*}S^{(k)}_{P},\widehat{M}_{S^{(k)}_{P}}\times S^{(k)}_{P})$
symplectomorphic to

$\displaystyle(\operatorname{Nbd}((h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}|_{\operatorname{Core}(H)})^{-1}(S^{(k)}_{P})),(h_{\Psi}\circ\operatorname{ret}\circ\operatorname{Log}|_{\operatorname{Core}(H)})^{-1}(S^{(k)}_{P}))$

for any $k$-stratum $S^{(k)}_{P}$ of $\Phi$ adjacent to a $0$-stratum $P$. By
construction they glue to yield a symplectomorphism of pairs

$\displaystyle(\widetilde{\mathbf{W}}(\Phi),\widetilde{\mathbb{L}}(\Phi))\hookrightarrow(H,\operatorname{Core}(H))$

over the pair $(\tilde{\pi}(\widetilde{\mathbf{W}}(\Phi)),\Phi)$. The same
modification as in the proof of Theorem 1.3 yields compatible fibrations over
$\Psi$. ∎

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# Delensing of Cosmic Microwave Background Polarization with machine learning

Ye-Peng Yan  Department of Astronomy, Beijing Normal University, Beijing
100875, China<EMAIL_ADDRESS>Guo-Jian Wang  School of Chemistry and
Physics, University of KwaZulu-Natal, Westville Campus, Private Bag X54001
Durban, 4000, South Africa  NAOC-UKZN Computational Astrophysics Centre
(NUCAC), University of KwaZulu-Natal, Durban, 4000, South Africa  Si-Yu Li
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Science, P. O. Box 918-3 Beijing 100049, People’s Republic
of China  Jun-Qing Xia  Department of Astronomy, Beijing Normal University,
Beijing 100875, China<EMAIL_ADDRESS>
###### Abstract

Primordial B-mode detection is one of the main goals of next-generation cosmic
microwave background (CMB) experiments. Primordial B-modes are a unique
signature of primordial gravitational waves (PGWs). However, the gravitational
interaction of CMB photons with large-scale structures will distort the
primordial E modes, adding a lensing B-mode component to the primordial B-mode
signal. Removing the lensing effect (‘delensing’) from observed CMB
polarization maps will be necessary to improve the constraint of PGWs and
obtain a primordial E-mode signal. Here, we introduce a deep convolutional
neural network model named multi-input multi-output U-net (MIMO-UNet) to
perform CMB delensing. The networks are trained on simulated CMB maps with
size $20^{\circ}\times 20^{\circ}$. We first use MIMO-UNet to reconstruct the
unlensing CMB polarization ($Q$ and $U$) maps from observed CMB maps. The
recovered E-mode power spectrum exhibits excellent agreement with the
primordial EE power spectrum. The recovery of the primordial B-mode power
spectrum for noise levels of 0, 1, and 2 $\mu$K-arcmin is greater than 98% at
the angular scale of $\ell<150$. We additionally reconstruct the lensing B map
from observed CMB maps. The recovery of the lensing B-mode power spectrum is
greater than roughly 99% at the scales of $\ell>200$. We delens observed
B-mode power spectrum by subtracting reconstructed lensing B-mode spectrum.
The recovery of tensor B-mode power spectrum for noise levels of 0, 1, 2
$\mu$K-arcmin is greater than 98 % at the angular scales of $\ell<120$. Even
at $\ell=160$, the recovery of tensor B-mode power spectrum is still around 71
%.

Cosmic microwave background radiation (322); Observational cosmology (1146);
Convolutional neural networks (1938)

##  1 INTRODUCTION

Inflationary models generically predict the existence of a stochastic
background of gravitational waves (i.e., primordial gravitational waves
(PGWs)) (Kamionkowski & Kovetz, 2016). The tensor-to-scalar ratio $r$, which
parameterize the PGW’s amplitude, is related to the energy scale at which
inflation occurred. Because of this, it is anticipated that the detection of
PGWs will reveal the physics of the universe’s very early stages. Fortunately,
these PGWs imprint a unique signature on the polarized anisotropies of the
cosmic microwave background (CMB) (Kamionkowski et al., 1997; Zaldarriaga &
Seljak, 1997) in the form of curl-like patterns B-modes, which makes it
possible to extract a signal of the PGW from B-modes of the CMB polarization.
To constrain the tensor-to-scalar ratio with high precision (Abazajian et al.,
2022), several next-generation CMB experiments with multi-frequency coverage
and very high sensitivity, such as the CMB-S4 project (Abazajian et al.,
2019), LiteBIRD satellite (Hazumi et al., 2019), and AliCPT (Li et al., 2017),
have been proposed or are under construction. However, PGWs are not the only
source of B-modes. Gravitational lensing of the CMB is a serious obstacle to
the search for PGWs. Gravitational lensing of the CMB arises from the
deflection of CMB photons as they pass through the matter distribution between
the surface of the last scattering and us, which results in a subtle remapping
of the temperature and polarization anisotropies (Lewis & Challinor, 2006).
Gravitational lensing can convert a small part of E-modes polarization into
B-modes polarization, which acts as a source of confusion in searches for
PGWs. Furthermore, lensing also smooths the acoustic peaks of the CMB power
spectra and produces non-stationary statistics of CMB fluctuations
(Zaldarriaga & Seljak, 1998; Lewis et al., 2001; Hotinli et al., 2022). In
order to improve constraints on the tensor-to-scalar ratio, we must remove the
effects of lensing (delensing) from observed CMB. Additionally, delensing of
observed CMB maps can sharpen acoustic peaks of T and E modes, and tighten
cosmological parameter constraints (Green et al., 2017).

CMB delensing methods have been investigated for many years (Knox & Song,
2002; Kesden et al., 2002; Seljak & Hirata, 2004; Smith et al., 2012; Larsen
et al., 2016; Carron et al., 2017; Millea et al., 2019, 2020; Sushovan Chandra
& Souradeep, 2021). Internal delensing methods require the subtraction of a
template of the lensing B-mode constructed from observed E-modes and a tracer
of the mass distribution that lensed the CMB. External delensing uses external
mass tracers which correlate with the CMB lensing signal, such as the cosmic
infrared background (CIB) (Simard et al., 2015; Sherwin & Schmittfull, 2015;
Yu et al., 2017), radio and optical galaxies (Namikawa et al., 2016; Manzotti,
2018), and intensity maps of high-redshift line emissions(Sigurdson & Cooray,
2005; Karkare, 2019). Some of these delensing methods have already been
demonstrated on data (Carron et al., 2017; Manzotti et al., 2017; Planck
Collaboration et al., 2020; Adachi et al., 2020; Han et al., 2021; BICEP/Keck
Collaboration et al., 2021), and have been used to constraint tensor-to-scalar
ratio. The current best constraints on the tensor-to-scalar ratio are $r<0.06$
at $95\%$ confidence obtained from the BICEP2/Keck Array (BICEP2 Collaboration
et al., 2018) or $r<0.036$ at $95\%$ (Ade et al., 2021) based on the BICEP2,
Keck Array, and BICEP3 CMB polarization experiments.

Figure 1: The architecture of the MIMO-UNet. The input images are observed CMB
(Q and U) maps, which is denoted by $O_{1}$. The $O_{2}$ and $O_{3}$ are down-
sample of $O1$ using the interpolation method. The sizes (${\rm(c\times
h\times w)=(channel\times height\times width})$) of each module’s feature
images are shown in figure. The input images ($O_{1},O_{2},O_{3}$) pass
through the convolution layers of related levels, and finally the three output
images ($D_{1}$, $D_{2}$, $D_{3}$) are obtained. Figure 2: The structures of
sub-modules of the MIMO-UNet. Conv-3 and Conv-1 denote convolutional layers
with $3\times 3$ and $1\times 1$ filters, respectively. SELU and $\bigotimes$
represent Scaled Exponential Linear Unit (SELU) function and multiplication,
respectively.

With the remarkable progress of computer science in recent years, machine
learning (ML) has demonstrated exceptional capabilities in the image
processing field, and it has been progressively used to CMB data processing.
For example, the methods of ML have been applied successfully to the CMB
component separation (Petroff et al., 2020; Wang et al., 2022; Casas et al.,
2022; Yan et al., 2023). In addition, Refs. (Caldeira et al., 2019; Guzman &
Meyers, 2021; Li et al., 2022; Heinrich et al., 2022; Guzman & Meyers, 2022)
also employ the ML approach for the reconstruction of CMB lensing and cosmic
polarization rotation maps. Caldeira et al. (2019) also uses ML method to
derive delensing E map for observed CMB Q/U maps. However, the delensing EE
power spectrum obviously deviates from true EE at the angular scales of
$\ell>1000$. Guzman & Meyers (2021) and Heinrich et al. (2022) improve the
performance in delensing E map, but the delensing EE power spectrum also
deviates from true EE with $\ell$ increases at the angular scales of
$\ell>1800$. As an improvement and development, in this work, we introduce a
convolutional neural network (CNN) to perform CMB delensing.

This paper is organized as follows. In Section 2, we describe our network
model as well as the simulated data sets. In Section 3, we present the results
of reconstructing the unlensing CMB ($Q$ and $U$) maps. In Section 4, we
present the results of reconstructing the lensing B map. In Section 5
discussions are presented. Finally, we conclude in Section 6.

##  2 Methodology

The deep learning architecture we use in this work, as well as the details of
the data pipeline and network training, will all be covered in this section.

###  2.1 Network architecture

Machine learning has demonstrated exceptional capabilities in the field of
image processing in recent years. The convolutional neural network (CNN) is
the most widely used type of feed-forward neural network in the tasks of image
processing (Mehta et al., 2019). The CNN’s core structure is the convolutional
layer, which computes the convolution of the input image with a bank of
filters (kernels) whose values are parameters to be learned. Each
convolutional layer accepts the image from the former layer as an input,
calculate the convolution of the input image, and finally gives output to the
next layer after applying a nonlinear activation function. The convolutional
layer generally has three hyper-parameters: the number of output channels (the
number of convolutional filters), stride, and amount of zero padding. The
output size of a convolutional layer can be controlled by setting these three
hyper-parameters. Here, the stride is defined as the distance in pixels
between the centers of adjacent filters.

In this work, we use the multi-input multi-output U-net (MIMO-UNet) (Cho et
al., 2021), an excellent network architecture for image deblurring tasks, to
perform the CMB delensing. The MIMO-UNet takes multi-scale input images to
ease the difficulty of training. As shown in Figure 1, MIMO-UNet model
consists of three levels. To obtain the multi-scale images ($O_{2}$, $O_{3}$),
we first perform down sampling for input image ($O_{1}$) using the
interpolation method, then take three scale images ($O_{1}$, $O_{2}$, $O_{3}$)
as an input of corresponding levels.

In the first level, we use ${\rm EB_{1}}$ block, which consists of a
convolutional layer and a residual block, to extract the features from the
input image $O_{1}$. This convolution layer is used to increase the number of
channel, and the residual block is used to extract the features. The residual
block consists of many residual layers as shown in panel (c) of Figure 2, and
we set the number of residual layers to 16 for a residual block in this work.
Then we use a shallow convolutional module (SCM) as shown in panel (a) to
extract the features from downsampled images ($O_{2}$ and $O_{3}$). The SCM
has two stacks of $3\times 3$ and $1\times 1$ convolutional layers. We
concatenate the features from the last $1\times 1$ layer with the input of
SCM, and further refine the concatenated features using an additional $1\times
1$ convolutional layer. The output of ${\rm EB_{1}}$ reduces the size by half
through a convolutional layer and is fed to the feature attention module (FAM)
as shown in panel (b) of Figure 2, and the FAM performs the element-wise
multiplied of this input and output of ${\rm SCM_{2}}$, and then the
multiplied features are passed through a $3\times 3$ convolutional layers. The
output of FAM is fed to residual block to further refine features. Finally,
the output of ${\rm EB_{2}}$ can be obtained. Similarly, the output of ${\rm
EB_{2}}$ and output of ${\rm SCM_{2}}$ are fed to ${\rm EB_{3}}$.

The outputs of ${\rm EB_{2}}$ and ${\rm EB_{3}}$ perform the up sample using
the interpolation method, and are fed to ${\rm AFF_{1}}$ together with the
output of ${\rm EB_{1}}$. The asymmetric feature fusion (AFF) module as shown
in panel (d) of Figure 2 allows information flow from different scales feature
maps. Similarly, The outputs of ${\rm EB_{2}}$, ${\rm EB_{3}}$, and ${\rm
EB_{1}}$ are also fed to ${\rm AFF_{2}}$. The output of the AFF is delivered
to its corresponding DB. ${\rm DB_{3}}$ consists of a residual block and
convolution transpose layer, which is used to double the sizes of the feature
maps. The output of ${\rm DB_{3}}$ and ${\rm AFF_{2}}$ be concatenated and fed
to ${\rm DB_{2}}$. Finally, the output of ${\rm DB_{2}}$ and ${\rm AFF_{1}}$
be concatenated and fed to ${\rm DB_{1}}$, and the delensing CMB Q and U maps
can be obtained. In addition, the second-level and third-level DBs also output
images before convolution transpose layer. The MIMO-UNet model is built in
PyTorch111https://pytorch.org/ environment, which is an open-source optimized
tensor library for deep learning.

The network model consists of a stack of non-linear parameters, and these
parameters can be optimized by minimizing a loss function when the network is
trained on a training set. We assume that the training set has $S$ pairs of
samples $\\{x_{i},y_{i}\\}_{i=1}^{S}$. Here, $x$ is the observed CMB Q and U
maps, and $y$ represents the corresponding ground truth of CMB maps. Our loss
function defines as

$\displaystyle\mathcal{L}=\sum^{3}_{k=1}(\mathcal{L}_{{\rm
LAD},k}+\beta\mathcal{L}_{{\rm FFT},k}),$ (1)

where $\mathcal{L}_{\rm LAD}$ is the least absolute deviation (LAD, also
called L1 loss), $\mathcal{L}_{\rm FFT}$ is the Fourier space loss, $k$ is the
number of levels of network, and $\beta$ is a coefficient representing the
contribution of $\mathcal{L}_{\rm FFT}$ to the total loss, and we set
$\beta=1$ throughout the paper. The L1 loss has form

$\displaystyle\mathcal{L}_{\rm LAD}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\left[\frac{1}{WH}\sum_{w=1}^{W}\sum_{h=1}^{H}(|I^{n}_{w,h}-y^{n}_{w,h}|)\right],$
(2)

where $N$ is the batch size, $H$ ($W$) is the height (width) of the images in
pixels, and $I=f(x)$ ($f(\cdot)$ is the network model) is the predicted image.
The form of $\mathcal{L}_{\rm FFT}$ is defined as

$\displaystyle\mathcal{L}_{\rm FFT}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\left[\frac{1}{WH}\sum_{w=1}^{W}\sum_{h=1}^{H}(|A_{\rm
F}(I^{n}_{w,h})-A_{\rm F}(y^{n}_{w,h})|)\right],$ (3)

where $A_{\rm F}$ is the amplitude of FFT, which has the form of

$\displaystyle A_{\rm F}(I)$ $\displaystyle=\sqrt{Re[{\rm FFT}(I)]^{2}+Im[{\rm
FFT}(I)]^{2}},$ (4)

where $Re[\cdot]$ and $Im[\cdot]$ denote the real and imaginary part,
respectively.

Figure 3: Examples of the observed and target CMB Q/U maps. The residual maps
are shown in the bottom row.

###  2.2 Data pipeline and network training

As a supervised machine learning technique, the CNN method requires a training
data set with values that are already known to be truth. The training data set
is based on a simulation of a flat sky. We use the
CAMB222https://github.com/cmbant/CAMB package to calculate the primordial CMB
and lensing potential power spectra (Lewis et al., 2001). Here, we consider a
standard $\Lambda$ cold dark matter model with parameters of
($H_{0},\Omega_{b}h^{2},\Omega_{c}h^{2},\tau,A_{s},n_{s}$), and their best-fit
value and standard deviation can be obtained from the Planck 2015 data (Planck
Collaboration et al., 2016). Additionally, we also take into account the
primordial tensor perturbation and set the tensor-to-scalar ratio $r$ to 0.05.
In this work, in order to get closer to a realistic manner, the values of
cosmological parameters are treated as independent Gaussian random variables,
where the mean values and standard deviations are derived from the best-fit
values and standard deviation of Planck-2015 result.

Figure 4: Example of the delensing CMB Q maps for four noise levels (0, 0.5,
1, 2 $\mu$K-arcmin; left to right). The top row is the target maps without
lensing effect, but with noise and beam information. The middle row is the
predicted delening maps by the MIMO-UNet. The bottom row shows the residual
maps. Table 1: Mean Absolute Error (MAE) between Two Maps over Test Set. MAE
between lensing maps and unlensing maps
---
- | MAE for the Q maps | MAE for the U maps
- | ($\mu$K) | ($\mu$K)
- | $1.074\pm 0.108$ | $1.169\pm 0.114$
MAE between predicted delensing maps and target maps
Noise | MAE for the predicted Q maps | MAE for the predicted U maps
($\mu$K-arcmin) | ($\mu$K) | ($\mu$K)
$0$ | $0.133\pm 0.008$ | $0.151\pm 0.009$
$0.5$ | $0.175\pm 0.009$ | $0.194\pm 0.009$
$1$ | $0.286\pm 0.015$ | $0.312\pm 0.014$
$2$ | $0.325\pm 0.015$ | $0.360\pm 0.015$
MAE between predicted delensing maps and target maps for patch size of
$15^{\circ}\times 15^{\circ}$
Noise | MAE for the predicted Q maps | MAE for the predicted U maps
($\mu$K-arcmin) | ($\mu$K) | ($\mu$K)
$0$ | $0.098\pm 0.006$ | $0.103\pm 0.006$
$1$ | $0.274\pm 0.014$ | $0.286\pm 0.013$
MAE between predicted delensing maps and target maps for patch size of
$10^{\circ}\times 10^{\circ}$
Noise | MAE for the predicted Q maps | MAE for the predicted U maps
($\mu$K-arcmin) | ($\mu$K) | ($\mu$K)
$0$ | $0.075\pm 0.004$ | $0.068\pm 0.004$
$1$ | $0.331\pm 0.020$ | $0.330\pm 0.019$
MAE between reconstructed lensing B map and target lensing B map
Noise | MAE for reconstructed lensing B map
($\mu$K-arcmin) | ($\mu$K)
$0$ | $0.028\pm 0.001$
$1$ | $0.035\pm 0.001$
$2$ | $0.062\pm 0.001$

Based on the these theoretical power spectra from CAMB, we simulate two-
dimensional flat sky maps by using a modified version of
Orphics333https://github.com/msyriac/orphics and resunet-
cmb444https://github.com/EEmGuzman/resunet-cmb. The simulated flat sky maps
cover $20^{\circ}\times 20^{\circ}$ patch of sky with $192\times 192$ pixels.
In total, three different types of maps are generated ($Q^{\rm prim},U^{\rm
prim},\kappa$). Here, the ‘prim’ represents the primordial CMB maps and
$\kappa$ is the convergence map. These three types of maps can be used to
generate observed CMB maps ($Q^{\rm observed},U^{\rm observed}$), which act as
the network’s input, and training target maps ($Q^{\rm target},U^{\rm
target}$). In order to obtain the observed maps ($Q^{\rm observed},U^{\rm
observed}$), we firstly lens the primordial maps ($Q^{\rm prim},U^{\rm prim}$)
with the convergence ($\kappa$) map. Then, similar to Refs. (Caldeira et al.,
2019; Guzman & Meyers, 2021), we smooth these lensed maps with a Gaussian beam
size of FWHM= 1 acrmin. Finally, we add a noise map to these beam convolved
lensed ($Q$, $U$) maps. Here, we choose four noise levels: $0.0$
$\mu$K-arcmin, $0.5$ $\mu$K-arcmin, $1$ $\mu$K-arcmin, and $2$ $\mu$K-arcmin.
These instrumental specifications are CMB-S4 like experiment. To obtain
training target maps ($Q^{\rm target},U^{\rm target}$), the primordial maps
($Q^{\rm prim},U^{\rm prim}$) are directly smoothed with a Gaussian beam and
then the noise maps is added. It should be noted that target maps does not
implement lensing effect. Here, the noise and beam on these target maps are
identical to those on the matching observed maps. Examples of observed and
target CMB maps are shown in the Figure 3. For quantitative comparison, we use
the mean absolute error (MAE) with the following general formula:

$\begin{split}\sigma_{\rm
MAE}&=\frac{1}{N}\sum_{i}^{N}\left|X_{i}-Y_{i}\right|,\end{split}$ (5)

where $N$ is the number of pixels, $X$ and $Y$ represent the predicted and
target maps. The MAE between observed (lensing) and target CMB Q/U maps
(unlensing) is calculated. As shown in Table 1, the MAEs are: $1.074\pm 0.108\
\mu$K for the Q map and $1.169\pm 0.114\ \mu$K for the U map. Due to the
lensing effect, we can see that the residual maps retain a lot of information.

For each noise level, 50000 sets of four independent maps ($Q^{\rm
observed},U^{\rm observed},Q^{\rm target},U^{\rm target}$) are generated and
split into training, validation, and test sets with a ratio of 8:1:1. Each
noise level is trained using a separate network. In order to train the
network, we adopt the Adam (Kingma & Ba, 2014) as the optimizer and initially
set the learning rate to 0.001, which gradually decreases to $10^{-6}$ during
the iterations. In training, we run 60000 iterations with a batch size of 32.
We train our network model on two NVIDIA Quadro GV100 GPUs, and one network
model take $\sim$ 27 hours to train. Due to the lensing effect, we can see
that the residual maps retain a lot of information.

##  3 Delensing CMB polarization maps with the MIMO-Unet model

In this section, we present results for delenisng CMB ($Q,U$) maps using the
MIMO-Unet model. The delensing CMB maps using the network are denoted by
($Q^{\rm predicted}$, $U^{\rm predicted}$). The observed CMB maps ($Q^{\rm
observed}$, $U^{\rm observed}$) are used as input of network, and the desired
output is unlensing CMB maps ($Q^{\rm target}$, $U^{\rm target}$) with
instrument noise and beam. This means that our network aims to find the
mapping between the observed CMB maps and delensing CMB maps, and does not
deal with noise and instrumental beam.

Figure 5: The delensed EE (top panels) and BB (bottom panels) power spectra
for four noise levels (0, 0.5, 1, 2 $\mu$K-arcmin; left to right). $\hat{E}$
and $\hat{B}$ represent predicted $E$ and $B$ obtained from our network’s
output ($Q,U$) maps. $E$ and $B$ represent the target $E$ and $B$.
$D_{\ell}^{\rm\hat{E}E}$ (or $D_{\ell}^{\rm\hat{B}B}$) represents the cross-
spectrum of predicted $\hat{E}$ (or $\hat{B}$) and target $E$ (or $B$). The
length of each $\ell$ bin is set to be 20 here. Here,
$D_{\ell}=\ell(\ell+1)C_{\ell}/2\pi$. We adopt the standard deviation on the
test set as the error of the estimate.

###  3.1 Delensing results

Our delensing Q maps (U maps have similar results) on the test set are shown
in Figure 4 for the experiments of four levels of noise. From visual
inspection, it is hard to distinguish between the predicted Q maps by network
and target maps. The bottom panels show the difference between the predicted
maps and targets. In the case of low noise levels ($\leq 1\mu$K-arcmin),
residual maps have minimal information left, implying that the MIMO-UNet can
delens the Q maps very well. The residual maps have more information at 2
$\mu$K-arcmin noise levels than at low noise levels, yet the MIMO-UNet still
has good delening performance when compared to the residual maps in Figure 3.
Furthermore, we can see that all amplitudes of residual maps grow visibly as
noise levels increase, implying that delensing performance decreases as noise
level increases. In order to evaluate the performance of our network
quantitatively, we calculate the MAE between predicted delensing and target Q
maps shown in Figure 4: $0.134\ \mu$K, $0.178\ \mu$K, $0.282\ \mu$K, $0.322\
\mu$K for 0 $\mu$K-arcmin, 0.5 $\mu$K-arcmin, 1 $\mu$K-arcmin, and 2
$\mu$K-arcmin noise levels, respectively. In addition, as shown in Table 1,
the average MAEs over the testing sets for delensing Q maps are $0.133\pm
0.008\ \mu$K, $0.175\pm 0.009\ \mu$K, $0.286\pm 0.015\ \mu$K, and $0.325\pm
0.015\ \mu$K for 0 $\mu$K-arcmin, 0.5 $\mu$K-arcmin, 1 $\mu$K-arcmin, and 2
$\mu$K-arcmin noise levels, respectively. The MAE for delensing U maps are
also listed by Table 1. These quantitative comparisons also support our
conclusions.

To make these observations more precise, we compute the power spectrum of
delensing CMB maps and compare them to the primordial EE and BB spectra. In
our method, the output CMB Q/U maps from the MIMO-UNet still contain the
instrument noise. According to Wang et al. (2022) and Yan et al. (2023), it is
still very challenging to directly remove noise effects at the map level using
CNN methods. Therefore, in this network we leave the noise in the output maps
and try to suppress the noise effect at the power spectrum level. Inspired by
(Krachmalnicoff et al., 2022), we divide the whole dataset into two ”half-
split” (HS) maps which share the same signal but have uncorrelated noise, and
then calculate the cross-correlation power spectrum of these two HS maps. Due
to their uncorrelated noise, the noise effect almost disappears in their
cross-correlation power spectrum, but the CMB signal remains. It should be
noted that the noise of HS maps is enhanced by a factor of $\sqrt{2}$ relative
to the primordial noise level.

We use NaMaster555https://github.com/LSSTDESC/NaMaster (Alonso et al., 2019)
to calculate the E- and B-mode power spectra from the predicted Q and U maps
by the MIMO-UNet. Figure 5 shows the delensed EE and BB spectra after denoised
step for four noise levels, and these power spectra are averaged over the test
set. The reconstructed $EE$ spectra ($D_{\ell}^{\hat{E}\hat{E}}$) are very
consistent with primordial EE spectra ($D_{\ell}^{EE}$) across the entire
range of scales we consider. Adding noise to the input maps does not degrade
the recovery of EE power spectrum significantly at all the noise levels, which
is consistent with results of (Caldeira et al., 2019; Guzman & Meyers, 2021).
The cross-correlation spectra ($D_{\ell}^{\hat{E}E}$) of reconstructed
$\hat{E}$ and true $E$ is also calculated. We can see that
$D_{\ell}^{\hat{E}E}$ are very consistent with $D_{\ell}^{\hat{E}\hat{E}}$,
which could imply that our reconstructed E-mode signal is the true signal.

The recovery of BB spectra ($D_{\ell}^{\hat{B}\hat{B}}$) gets systematically
worse as $\ell$ increases. Specifically, the recovery of the B-mode power
spectrum is greater than about 98% of the primordial B-mode at the angular
scale of $\ell<150$, but it gradually deviates from the primordial BB
($D_{\ell}^{BB}$) as $\ell$ increases at the scales of $\ell>150$. In
particular, the recovered B-mode power spectrum significantly deviates from
the primordial BB at the scales of $\ell>200$. In fact, the lensing B-mode is
at least one order of magnitude higher than the primordial BB at the scales of
$\ell>200$, implying that recovering the primordial B-mode is extremely
challenging. Therefore, we think that the recovered B-mode power spectrum by
our network should be dominant by lensing B-mode at the scales of $\ell>200$
because our network does not completely remove the lensing effect at these
scales. A more detailed analysis will be presented in section 3.2. The cross-
correlation spectra ($D_{\ell}^{\hat{B}B}$) of reconstructed $\hat{B}$ and
true $B$ is also calculated. We can see that $D_{\ell}^{\hat{B}B}$ are
consistent with $D_{\ell}^{\hat{B}\hat{B}}$ at the scales of $\ell<140$, but
$D_{\ell}^{\hat{B}B}$ gradually deviates from the $D_{\ell}^{\hat{B}\hat{B}}$
at the scales of $\ell>140$. In particular, at the scales of $\ell>200$,
$D_{\ell}^{\hat{B}B}$ is about 60-50% of $D_{\ell}^{BB}$. These could suggest
that our recovered BB spectra ($D_{\ell}^{\hat{B}\hat{B}}$) are true
primordial B-mode signals at the scales of $\ell>140$, but the primordial
B-mode signal contained in $D_{\ell}^{\hat{B}\hat{B}}$ will gradually lose as
$\ell$ increases.

###  3.2 Null test

Now, we need to check that the network encodes a sensible mapping of the input
observed maps to delensing maps, i.e., the reconstructed BB spectrum by the
network model is due to the presence of the real tensor BB rather than an
artifact of the network and it is the real tensor BB rather than a reduced
lensing BB. In addition, we also wish to understand why the reconstructed
tensor BB deviates from the primordial BB spectrum at the scales of
$\ell>150$. Based on these considerations, two null tests are performed.

Figure 6: Null test for experiments of 0 and 1 $\mu$K-arcmin noise levels. Top
panel: null test 1 represents that we feed unlensed versions of ($Q$,$U$) maps
with $r=0$ to MIMO-Unet model trained on the observed ($Q^{\rm
observed},U^{\rm observed}$) maps. Bottom panel: null test 2 represents that
we feed lensed versions of ($Q$, $U$) maps with $r=0$ to the network model
trained on the observed ($Q^{\rm observed},U^{\rm observed}$) maps.
$D_{\ell}^{\rm Predicted\ BB}$ and $D_{\ell}^{\rm Null\ test\ BB}$ are power
spectrum of normal network output shown in Figure 5 and power spectrum from
null test output, respectively.

For the first null test (null test 1), we feed the unlensed versions of
($Q$,$U$) maps with no tensor-to-scalar ratio (i.e., $r=0$) into the network
model trained on the observed ($Q^{\rm observed},U^{\rm observed}$) maps.
Here, the noise and beam on these unlensed versions maps are the same as those
on observed ($Q^{\rm observed},U^{\rm observed}$) maps. We calculate the power
spectrum of the output ($Q^{\rm pred},U^{\rm pred}$) from this null test. The
upper panel of Figure 6 shows the predicted power spectrum from this null test
for noiseless and 1 $\mu$K-arcmin noise experiments. The predicted BB spectrum
for 1 $\mu$K-arcmin noise is denoised by cross-correlation of two HS maps. We
can see that the predicted BB spectra from the null test look like the noise
signals, and predicted $D_{\ell}^{BB}$ for 1 $\mu$K-arcmin noise level is
higher than that of noiseless experiment. These imply that the output ($Q^{\rm
pred},U^{\rm pred}$) could be the random noise when we input the unlensed
(Q,U) maps with $r=0$. Furthermore, the amplitude of this output noise depends
on the input noise level.

Figure 7: Example of reconstructing lensing B maps for three noise levels (0,
1, 2 $\mu$K-arcmin; left to right). The top row is the target lensing B maps
($B_{\rm lensing}^{\rm tar}$, that is, observed lensing B map $B^{\rm obs\
lensing}$). The middle row is the predicted lensing B maps ($B_{\rm pred}^{\rm
tar}$) by the MIMO-UNet. The bottom row shows the related residual maps.

Next, we perform the second null test (null test 2) and feed lensed versions
of ($Q$, $U$) maps with $r=0$ to the network model trained on the observed
($Q^{\rm observed},U^{\rm observed}$) maps. The noise and beam on these lensed
versions maps are the same as those on observed ($Q^{\rm observed},U^{\rm
observed}$) maps. The bottom panel of Figure 6 shows that the power spectra of
the output ($Q^{\rm pred},U^{\rm pred}$) from this null test for noiseless and
1 $\mu$K-arcmin noise experiments. We can see that the predicted BB spectra
from null test 2 still look like the noise signals, which suggests that the
reconstructed BB spectrum by the network model is from the real tensor BB. In
addition, since the input map contains lensing effect, BB spectra from null
test 2 are higher than those null test 1. We can understand this as: our
network has not completely removed the lensing effect, that is, there is still
a small amount of lensing on the output map of our network, which can also be
observed on the residual maps of Figure 4. These small amounts of lensing on
the output map will contribute to the BB spectrum ($D_{\ell}^{BB,{\rm reduced\
lensing}}$). As a result, the predicted BB spectrum ($D_{\ell}^{BB,{\rm
predicted}}$) from null test 2 can be modeled as

$\displaystyle D_{\ell}^{BB,{\rm predicted}}=D_{\ell}^{BB,{\rm reduced\
lensing}}+N_{\ell},$ (6)

where $N_{\ell}$ is the noise spectrum. We also plot the delensing BB spectrum
by network from observed ($Q^{\rm observed},U^{\rm observed}$) maps in the
bottom panel of Figure 6. We see that predicted BB spectra by our network are
consistent with the results of null test 2 at the angular scales of $\ell>300$
for the noiseless experiment and $\ell>250$ for 1 $\mu$K-arcmin noise
experiment. These could suggest that the reconstructed tensor BB deviates from
the primordial BB spectrum at the scales of $\ell>150$ as shown in Figure 5
due to the contributions of $D_{\ell}^{BB,{\rm reduced\ lensing}}$ and
$N_{\ell}$.

##  4 Reconstruct lensing B map

###  4.1 Reconstructed results

In section 3.1, we use the CNN to delens Q and U maps. Inspired by Manzotti et
al. (2017), we now try to reconstruct the lensing B mode using the CNN, and we
delens observed B-mode power spectrum by subtracting this reconstructed
lensing B-mode power spectrum.

Figure 8: The recovered BB power spectra for experiments of three noise levels
(0, 1, 2 $\mu$K-arcmin; left to right). $B$ and $\hat{B}$ represent the true
lensing B map and predicted lensing B map by the network. $C_{\ell}^{\rm
BB,lensing}$ and $C_{\ell}^{\rm BB,tensor}$ are true lensing BB spectrum and
primordial tensor BB spectrum. $C_{\ell}^{B\hat{B}}$ is cross-correlation
between $B$ and $\hat{B}$. The $C_{\ell}^{\hat{B}\hat{B}}$ is predicted
lensing BB spectrum by network. $C_{\ell}^{\rm BB,input}$ is network input BB
spectrum from observed CMB $Q$ and $U$ maps. The power spectra are averaged
over the test set, and we adopt the standard deviation on the test set as the
error of the estimate of the tensor BB spectrum.

Similar to the simulation process in Section 2.2, we simulate five different
types of maps in total, ($Q^{\rm prim}$, $U^{\rm prim}$, $Q^{\rm prim,r=0}$,
$U^{\rm prim,r=0}$, $\kappa$). Here, the ‘prim’ and ‘prim,r=0’ represent the
primordial CMB maps with $r=0.05$ and primordial CMB maps with $r=0$. $\kappa$
is the convergence map. Based on these five types of maps, the observed maps
($Q^{\rm observed}$, $U^{\rm observed}$, $Q^{\rm observed,r=0}$, $U^{\rm
observed,r=0}$) and training target maps of the network can be generated.
Here, the ‘observed’ and ‘observed,r=0’ represent the observed CMB maps with
r=0.05 and observed CMB maps with r=0. To obtain observed maps, we firstly
lens the primordial maps (($Q^{\rm prim},U^{\rm prim}$) or ($Q^{\rm
prim,r=0}$, $U^{\rm prim,r=0}$)) with the convergence ($\kappa$) map. Then, we
smooth these lensed maps with a Gaussian beam size of FWHM= 1 acrmin. Finally,
we add a noise map to these beam convolved lensed ($Q$, $U$) maps. Here, we
choose three levels of noise: $0.0$ $\mu$K-arcmin, $1$ $\mu$K-arcmin, and $2$
$\mu$K-arcmin. Note that the observed maps with $r=0.05$ ($Q^{\rm observed}$,
$U^{\rm observed}$) take as input of network. In this section, the desired
output of the network is observed lensing B map ($B^{\rm obs\ lensing}$).
Here, $B^{\rm obs\ lensing}$ map is generated using observed CMB maps with
$r=0$ ($Q^{\rm observed,r=0}$, $U^{\rm observed,r=0}$). This means that the
network is designed to reconstruct lensing B map from observed ($Q^{\rm
observed}$, $U^{\rm observed}$) maps. For each noise level, 50000 sets of
three independent maps ($Q^{\rm observed}$, $U^{\rm observed}$, $B^{\rm obs\
lensing}$) are generated and split into training, validation, and test sets
with a ratio of 8:1:1. Each noise level is trained using a separate network.

Figure 7 shows the reconstructed lensing B maps for three noise levels (0, 1,
2 $\mu$K-arcmin) on the test set. From visual inspection, we see that there is
little information left on the residual map, implying that the network can
reconstruct lensing B map fairly well. However, as the levels of noise
increasing, all the amplitudes of residual maps increase. This implies that
the performance of reconstructing lensing B map by network depends on the
noise level of the observed map. The MAEs between predicted lensing and target
lensing B maps as shown in Figure 7 is calculated for the experiments of three
levels of noise: $0.029\ \mu$K, $0.036\ \mu$K, and $0.061\mu$K for 0
$\mu$K-arcmin, 1 $\mu$K-arcmin, 2 $\mu$K-arcmin noise levels, respectively. In
addition, as shown in Table 1, the average MAEs over the testing sets are
$0.028\pm 0.001\ \mu$K, $0.035\pm 0.001\ \mu$K, and $0.062\pm 0.001\ \mu$K for
0 $\mu$K-arcmin, 1 $\mu$K-arcmin, 2 $\mu$K-arcmin noise levels, respectively.
These quantitative comparisons also support our conclusions.

Next, we present the power spectrum of the reconstructed lensing B map at each
noise level. Figure 8 shows the power spectra of reconstructed lensing B map
for three noise levels: noiseless, 1 $\mu$K-arcmin, 2 $\mu$K-arcmin. These
power spectra are the mean result over the test set. Note that the
reconstructed power spectra by the network are denoised by the cross-
correlation of HS maps. At the angular scales of $\ell>200$, the reconstructed
lensing BB power spectra ($C^{\hat{B}\hat{B}}_{\ell}$) by our network are
quite consistent with the truth lensing BB spectra ($C^{BB,{\rm
lensing}}_{\ell}$). The recoveries of lensing BB power spectra are greater
than about 99% and adding noise to input maps does not degrade lensing-B mode
recovery significantly from the noiseless case. However, the reconstructed
lensing BB ($C^{\hat{B}\hat{B}}_{\ell}$) slightly deviates from the truth
lensing BB at the angular scales of $\ell<200$, which could be affected by
tensor B-mode. The recovery of the lenging BB at the angular scales of
$\ell<200$ visibly degrades when noise is added to the input maps. This
implies that the noise has a negative impact on lensing BB spectra
reconstruction, which is consistent with the obtained results of the
reconstructed lensing B map in Figure 7.

We also plot the cross-correlation spectrum ($C^{B\hat{B}}_{\ell}$) between
the reconstructed lensing B map and true lensing B map in Figure 8. We see
that reconstructed BB spectra are consistent with $C^{B\hat{B}}_{\ell}$ for
all noise levels, which implies that our reconstructed lensing B map is a true
lensing B mode signal.

Next, we delens our observed BB power spectrum by subtracting the
reconstructed lensing BB spectrum. This ‘spectrum difference’ is defined as

$\displaystyle\Delta C^{\hat{B}\hat{B}}_{\ell}\equiv$ $\displaystyle
C^{BB,{\rm obs}}_{\ell}-C^{\hat{B}\hat{B}}_{\ell}.$ (7)

We use this $\Delta C^{\hat{B}\hat{B}}_{\ell}$ as the estimate of the tensor
BB spectrum. We also plot the $\Delta C^{\hat{B}\hat{B}}_{\ell}$ (yellow
dotted line) for four noise levels in Figure 8. We see that $\Delta
C^{\hat{B}\hat{B}}_{\ell}$ are basically consistent with tensor BB spectrum
($C^{BB,{\rm tensor}}$) at the angular scales of $\ell<180$, and adding noise
to input maps does not degrade tensor-B mode recovery significantly. However,
$\Delta C^{BB}_{\ell}$ deviates from the truth tensor BB at the angular scales
of $\ell>180$. More specifically, the B-mode power spectrum recovery at noise
levels of 0, 1, 2 $\mu$K-arcmin is greater than about 98% at the scales of
$\ell<120$. Tensor BB power spectrum recovery is around 71% at $\ell=160$. We
also plot the difference between the observed BB power spectrum and cross-
correlation spectrum in Figure 8, which is defined as

$\displaystyle\Delta C^{B\hat{B}}_{\ell}\equiv$ $\displaystyle C^{BB,{\rm
obs}}_{\ell}-C^{B\hat{B}}_{\ell}.$ (8)

We see that $\Delta C^{B\hat{B}}_{\ell}$ are consistent with $\Delta
C^{\hat{B}\hat{B}}_{\ell}$ at the angular scales of $\ell<200$ for all noise
levels.

We note that the errors of the recovered tensor B-mode power spectrum increase
sharply at the angular scales of $\ell>220$ for the experiment of 1
$\mu$K-arcmin noise level and $\ell>190$ for the experiment of 2 $\mu$K-arcmin
noise level, while error for the noiseless experiment does not increase
significantly with the increase of $\ell$. This implies that the increase in
error could be due to the presence of noise. In fact, the reconstructed map by
network model contains the instrumental noise, and we always hope that we can
use the cross-correlation technique to suppress the instrumental noise effect
on the power spectrum. However, it is difficult to suppress noise on small
scales because the tensor B-mode signal is very weak. This could lead to an
increase in the error of the recovered tensor B-mode power spectrum at the
small scales. Specifically, the reconstructed $\hat{B}$ map can be written as

$\displaystyle\hat{B}_{\ell}=B_{\ell}+N_{l},$ (9)

where $N_{l}$ and $B_{\ell}$ are noise and true B-mode field. The cross-
correlation of the two reconstructed HS maps, $\hat{B}^{1}$ and $\hat{B}^{2}$,
can be calculated as

$\displaystyle\langle\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\rangle=$
$\displaystyle\langle B_{\ell}B_{\ell}\rangle+\langle
B_{\ell}N^{1}_{\ell}\rangle+\langle B_{\ell}N^{2}_{\ell}\rangle+\langle
N^{1}_{\ell}N^{2}_{\ell}\rangle.$ (10)

Here, $\langle\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\rangle$ is an unbiased
estimate of $\langle B_{\ell}B_{\ell}\rangle$. We can calculate the variance
of $\langle\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\rangle$ as

$\displaystyle\langle(\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}-B_{\ell}B_{\ell})^{2}\rangle=$
$\displaystyle\langle\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\rangle+(C_{\ell}^{BB})^{2}$
(11) $\displaystyle-2\langle\hat{B}_{\ell}^{1}\hat{B}_{\ell}^{2}\rangle
C_{\ell}^{BB}$ (12) $\displaystyle=$ $\displaystyle
C_{\ell}^{BB}(N_{\ell}^{1,1}+N_{\ell}^{2,2})+N_{\ell}^{1,1}N_{\ell}^{2,2},$
(13)

where $N_{\ell}^{1,1}$ and $N_{\ell}^{2,2}$ are auto-correlation spectra of
noise $N^{1}$ and $N^{2}$. $N_{\ell}^{1,1}N_{\ell}^{2,2}$ is the main source
of variance of cross-correlation at about scale of $\ell>200$ for noise level
of 1 $\mu$K-arcmin and $\ell>160$ for noise level of 2 $\mu$K-arcmin.
Therefore, the errors of the recovered tensor B-mode power spectrum after
denoised step using cross-correlation of HS maps increase sharply at small
scale.

Figure 9: Null test for experiments of 0 and 1 $\mu$K-arcmin noise levels. Top
panel: null test 1 represents that we feed unlensed versions of ($Q$,$U$) maps
with $r=0$ to MIMO-Unet model trained on the observed ($Q^{\rm
observed},U^{\rm observed}$) maps. Bottom panel: null test 2 represents that
we feed unlensed versions of ($Q,U$) maps with $r=0.05$ to the network model
trained on the observed ($Q^{\rm observed},U^{\rm observed}$) maps.
$C_{\ell}^{\rm BB,lensing}$ and $C_{\ell}^{\rm BB,tensor}$ are true lensing BB
spectrum and primordial tensor BB spectrum. $C_{\ell}^{\rm Null\ test\ BB}$ is
the power spectrum from null test output.

###  4.2 Null test

Similar to section 3.2, to verify that the network encodes a sensible mapping
of the input observed map to lensing B map, we apply two null tests.

For the first null test (null test 1), we feed unlensed versions of ($Q,U$)
maps with no tensor-to-scalar ratio (i.e., $r=0$) to the network model trained
on the observed ($Q^{\rm observed}$, $U^{\rm observed}$) maps. Here, the noise
and beam on these unlensed versions maps are the same as those on observed
($Q^{\rm observed},U^{\rm observed}$) maps. We consider two experiments with
noiseless and 1 $\mu$K-arcmin noise. The power spectrum of the output from
this null test is shown in the upper panel of Figure 9. We can see that our
reconstructed lensing BB power spectra from the null test have a small signal
and look like the noise signals. In addition, the reconstructed lensing BB
from the null test is one to three orders of magnitude smaller than the true
lensing power spectrum. This implies that our reconstructed lensing B map is
true lensing B signal, rather than producing artifacts from the training
process. In addition, our reconstructed lensing BB power spectra from the null
tests (i.e., noise signal from the trained network) are one to two orders of
magnitude smaller than the tensor BB spectrum, which means that this noise
signal from the network has little effect on the reconstructed tensor BB power
spectrum.

Figure 10: The delensed BB power spectra for experiments of three tensor-to-
scalar ratio levels ($r=0.005,0.01,0.05$; left to right). Predicted power
spectra by the network are denoised by the cross-correlation of HS maps and
averaged over the test set. $D_{\ell}^{\rm BB,tensor}$ and
$D_{\ell}^{\rm\hat{B}\hat{B}}$ are primordial tensor BB spectrum and predicted
BB spectrum by network. We set noise level to 1 $\mu$K-arcmin.

For the second null test, we feed unlensed versions of ($Q,U$) maps with
$r=0.05$ to the network model trained on the observed ($Q^{\rm observed}$,
$U^{\rm observed}$) maps. Here, the noise and beam on these unlensed versions
maps are the same as those on observed ($Q^{\rm observed},U^{\rm observed}$)
maps. The power spectrum of output maps from this null test for experiments of
noiseless and 1 $\mu$K-arcmin noise are shown in the bottom panel of Figure 9.
The predicted lensing BB power spectra from this null test also have a small
signal similar to noise and are one to two orders of magnitude smaller than
the true lensing power spectrum. In addition, this noise signal is higher than
noise signal from null test 1 because the tensor-to-scalar ratio is added to
the test set of the second null test. We also compare this noise signal from
null test 2 output to the tensor BB spectrum as shown in the bottom panel of
Figure 9. This noise signal is one to two orders of magnitude smaller than the
tensor BB spectrum at the angular scales of $\ell<100$. The difference between
the two signals gradually decreases at the angular scales of $100<\ell<300$,
and these two signals are almost identical at the angular scales of
$\ell>300$. These imply that the tensor B signal will have a negative impact
on the reconstruction of lensing B map. Our network cannot recognize the
tensor BB signal from the observed B mode signal (lensing B model plus tensor
B mode) at the angular scales of $\ell>200$ because the tensor BB is at least
one order of magnitude smaller than lensing BB. This suggests that our method
cannot accurately recover the tensor BB spectrum using E.q. 7 at the angular
scales of $\ell>200$. This conclusion has been demonstrated by our above
results as shown in Figure 8, where our recovered tensor BB spectrum using
E.q. 7 is higher than true tensor BB spectrum at the small angular scales of
$\ell>150$.

##  5 DISCUSSIONS

###  5.1 Variability of the tensor-to-scalar ratio

In section 3 and 4, we presented the results of delening CMB polarization maps
and reconstructing lensing B map based on the training set with a fixed
tensor-to-scalar ratio of 0.05. However, in real observation, we do not know
what is its tensor-to-scalar ratio. To enable the network to handle unknown
tensor-to-scalar ratio data, we now consider the variation of the tensor-to-
scalar ratio ($r$) on the training set. Similar to the simulation process
described in Section 2.2, we here set the tensor-to-scalar ratio to vary in
the range of $[0.0001,0.1]$ according to a uniform distribution. Then, 50000
sets of maps are independently generated and split into training and
validation sets with a ratio of 8:2. The training set is used to train MIMO-
Unet model. Therefore, during training, our network can see CMB data with
tensor-to-scalar ratios of $r\in[0.0001,0.1]$. This means that the trained
network model can handle unknown tensor-to-scalar ratios data in the range of
$[0.0001,0.1]$. To obtain the network predictions, we evaluate the trained
MIMO-Unet model on three sets of 5000 test maps, and these three test sets
have the fixed tensor-to-scalar ratio, $r_{\rm val}=0.005$, $r_{\rm
val}=0.01$, $r_{\rm val}=0.05$, respectively. Here, we choose these three
fixed tensor-to-scalar ratio as an example to evaluate our network.

Figure 11: The recovered BB power spectra for experiments of three tensor-to-
scalar ratio levels ($r=0.005,0.01,0.05$; left to right). $C_{\ell}^{\rm
BB,lensing}$ and $C_{\ell}^{\rm BB,tensor}$ are true lensing BB spectrum and
primordial tensor BB spectrum. $C_{\ell}^{\hat{B}\hat{B}}$ is predicted
lensing BB spectrum. $C_{\ell}^{\rm BB,input}$ is the network input BB power
spectrum derived from observed CMB $Q$ and $U$ maps. We set the noise level to
1 $\mu$K-arcmin.

We first apply the MIMO-Unet model to the task of delensing CMB polarization
maps similar to the Section 3. The observed CMB maps ($Q^{\rm observed}$,
$U^{\rm observed}$) are used as input, and the desired output is delensing CMB
maps ($Q^{\rm tar}$, $U^{\rm tar}$) with the instrument noise and beam. Figure
10 plots the delensed BB spectra of three test sets after denoised step. As a
discussion, we only show the 1 $\mu$K-arcmin noise level result; other noise
level results are similar to this one. The BB spectra are averaged over the
test sets. We see that the delens BB spectrum for the case of $r=0.05$ is
consistent with the tensor BB spectrum at the angular scales of $\ell<100$,
which is worse than the result shown in Figure 5 at $100<\ell<150$ where the
network is trained on a fixed $r=0.05$ training set. Therefore, the variation
of the tensor-to-scalar ratio on the training set will degrade slightly the
reconstructed performance of the network at the angular scales of
$100<\ell<150$. In addition, the reconstructed BB spectrum visibly degrades as
$r$ decreases. The tensor B-mode power spectrum recovery at the angular scales
of $\ell<100$ is about 77% for observed maps with r = 0.01 and 56 % for
observed maps with r = 0.005. This implies that it is still very challenging
for our network to reconstruct tensor BB spectrum with smaller $r$ through
directly delensing Q and U maps.

Next, we apply the MIMO-Unet model to the task of reconstructing lensing B map
similar to the section 4. The observed CMB maps ($Q^{\rm observed}$, $U^{\rm
observed}$) are used as input, and the desired output is the observed lensing
B map ($B^{\rm observed,lensing}$) with the instrument noise and beam. Figure
11 shows the power spectra of reconstructed lensing B map for three test sets
with $r=0.005$, $r=0.01$, $r=0.05$. We also only show the results of 1
$\mu$K-arcmin noise experiment. The reconstructed lensing BB power spectra are
the mean result over the testing sets and are denoised by the cross-
correlation of HS maps. For the case of $r=0.05$, we see that the
reconstructed lensing BB power spectrum is quite consistent with the truth
lensing BB spectra at $\ell>200$, and slightly deviates from the truth lensing
BB at the angular scales of $\ell<200$. We also plot the delensing BB power
spectrum using E.q. 7. We see that the delensing BB spectrum is basically
consistent with the tensor BB spectrum at the angular scales of $\ell<200$.
These results are consistent with the result shown in Figure 8 where the
network is trained on a fixed $r=0.05$ training set. This implies that the
variation of the tensor-to-scalar ratio on the training set will not
significantly degrade the reconstructed lensing B map performance. In
addition, our network can reconstruct lensing BB power spectrum for test sets
with $r=0.005$ and $r=0.01$. However, delensing BB power spectrum using E.q. 7
still degrades as $r$ decreases, which is consistent with the result above.
The tensor B-mode power spectrum recovery at the angular scales of $\ell<100$
is around 71% for observed maps with r = 0.01 and 56 % for observed maps with
r = 0.005.

###  5.2 Variability of size of sky patch

In section 3 and 4, we presented the results of delening CMB polarization maps
and reconstructing lensing B map using the $20^{\circ}\times 20^{\circ}$ sized
maps with $192\times 192$ pixels. Here, we investigate how the results depend
on the size of sky patches. We simulate the flat maps that cover
$15^{\circ}\times 15^{\circ}$ and $10^{\circ}\times 10^{\circ}$ patch of sky
with $192\times 192$ pixels as described in section 2.2 and 4.1. Considering
that we simulated the flat maps, we do not increase the area of the sky area.
As a discussion, we only show the results for 0 and 1 $\mu$K-arcmin noise
level.

We first apply the MIMO-Unet model to the task of delensing CMB polarization
maps similar to Section 3. The observed CMB maps ($Q^{\rm observed}$, $U^{\rm
observed}$) are used as input, and the desired output is delensing CMB maps
($Q^{\rm tar}$, $U^{\rm tar}$) with the instrument noise and beam. Figure 12
plots the delensed BB spectra for the experiments with two the sizes of sky
patches ($15^{\circ}\times 15^{\circ}$ and $10^{\circ}\times 10^{\circ}$). In
noiseless case, residual maps have minimal information left, implying that the
MIMO-UNet can cleanly delens the Q maps for observed maps with patch sizes of
$15^{\circ}\times 15^{\circ}$ and $10^{\circ}\times 10^{\circ}$. We calculate
the average MAE between predicted delensing and target maps. As shown in table
1, the average MAEs over test set from patch size of $15^{\circ}\times
15^{\circ}$ are $0.098\pm 0.006\ \mu$K and $0.103\pm 0.006\ \mu$K for
delensing Q maps and U maps. The the average MAEs from patch size of
$10^{\circ}\times 10^{\circ}$ are $0.075\pm 0.004\ \mu$K and $0.068\pm 0.004\
\mu$K for delensing Q maps and U maps. These values are slightly smaller than
the results from sizes sky patches $20^{\circ}\times 20^{\circ}$ for the
noiseless case. However, once the noise is present, the results are different
from the noiseless case. As shown in table 1, for the case of $1\mu$K-arcmin
noise, the average MAEs over the test set from patch size of $15^{\circ}\times
15^{\circ}$ are $0.274\pm 0.014\ \mu$K and $0.286\pm 0.013\ \mu$K for
delensing Q maps and U maps. These values are consistent with the results from
sizes sky patches $20^{\circ}\times 20^{\circ}$. The the average MAEs from
patch size of $10^{\circ}\times 10^{\circ}$ are $0.331\pm 0.020\ \mu$K and
$0.330\pm 0.019\ \mu$K for delensing Q maps and U maps. These values are
slightly larger than the results from patch size of $20^{\circ}\times
20^{\circ}$.

Figure 12: Example of the delensing CMB Q maps for two different sized sky
areas ($15^{\circ}\times 15^{\circ}$ and $10^{\circ}\times 10^{\circ}$) and
two different noise levels (0 and 1 $\mu$K-arcmin). The top row shows the
target maps without lensing effect, but with noise and beam information. The
middle row is the predicted delening maps by the MIMO-UNet. The bottom row
shows the residual maps. Figure 13: The power spectra of the delensed EE (top
panels) and BB (bottom panels) for two different sized sky areas
($15^{\circ}\times 15^{\circ}$ and $10^{\circ}\times 10^{\circ}$) and two
different noise levels (0 and 1 $\mu$K-arcmin). $\hat{E}$ and $\hat{B}$
represent predicted $E$ and $B$ obtained from our network’s output ($Q,U$)
maps. $E$ and $B$ represent the target $E$ and $B$. The length of each $\ell$
bin is set to be 20 here.

Figure 5 shows the delensed EE and BB spectra after denoised step for the
experiments with two the sizes of sky patches ($15^{\circ}\times 15^{\circ}$
and $10^{\circ}\times 10^{\circ}$). The reconstructed $EE$ spectra
($D_{\ell}^{\hat{E}\hat{E}}$) are very consistent with primordial EE spectra
($D_{\ell}^{EE}$) at $\ell<3000$. The B-mode power spectrum recovers more than
98% of the primordial B-mode at the angular scale of $\ell<150$, but it
gradually deviates from the primordial BB ($D_{\ell}^{BB}$) as $\ell$
increases at the scales of $\ell>150$. These results are consistent with
results of Section 3.1. These imply that our results are independent of sky
area size from the perspective of the recovered power spectrum.

Based on these analyses from recovered primordial (Q and U) maps and power
spectra, we can conclude that the size of the sky area has a slight effect on
the results, which is primarily seen in the recovered maps, but our network is
still able to handle effectively data with a variety of sky patch sizes. Here,
we do not show the dependence of the reconstructed lensing B-mode results
(refer to Section 4.1) on the size of sky patches, but we have checked that
these conclusions drawn in this section still stand.

We note that E-mode power spectrum recovery is greater than about 98% for the
angular scales of $\ell<2000$. At about $\ell=3000$, EE power spectrum
recovery is around 80%. This suggests that our network is very effective at
recovering the primordial EE power spectrum. For comparison with the results
of Guzman & Meyers (2021), we simulate flat maps that cover $5^{\circ}\times
5^{\circ}$ a patch of sky with $128\times 128$ pixels as described in section
2.2. This resolution of the map is consistent with Guzman & Meyers (2021). As
a discussion, we only consider the case of 1 $\mu$K-arcmin noise level. The
observed CMB maps ($Q^{\rm observed}$, $U^{\rm observed}$) are used as input,
and the desired output is delensing CMB maps ($Q^{\rm tar}$, $U^{\rm tar}$)
with the instrument noise and beam. We plot the delensed EE spectra after
denoised step as shown in Figure 14. We see that E-mode power spectrum
recovery is greater than about 98% for the angular scales of $\ell<2000$. At
about $\ell=3000$, EE power spectrum recovery is around 87%. This result is
better than that of Guzman & Meyers (2021). We do not show the B-mode power
spectrum recovery because it is similar to the Section 3 results.

Figure 14: The delensed EE power spectra for high-resolution CMB maps with
$5^{\circ}\times 5^{\circ}$ patch of sky and $128\times 128$ pixels. The true
spectrum and predicted spectrum represent the primordial EE spectrum and
delensed EE spectrum using the MIMO-UNet, respectively. We set noise level to
1 $\mu$K-arcmin.

###  5.3 The CNN Method

In this work, we demonstrated that the CNN can be used to recover unlensed CMB
maps and lensing B map in simulated data. There are some challenges still
present in the use of CNN method. The CNN method is to extract features from
images and pass these features to later layers for the further filtering out
of interesting signals. We find that our network is difficult to extract
tensor B-model signals of the small scales because the features of tensor
B-model map at small scale are very small. The large-scale features of the
tensor B-model are challenging to recover when the tensor-to-scalar ratio is
small $r<0.01$. In addition, the noise on the observed map will pollute the
features of interesting signals and will degrade the performance of the
network.

##  6 CONCLUSIONS

In this paper, we presented a machine-learning CMB polarization delensing
technique and demonstrated that the machine-learning can be used to delens CMB
polarization maps in simulated data. The networks were trained using
$20^{\circ}\times 20^{\circ}$ sized simulated CMB $Q$ and $U$ maps. We studied
two delensing methods. For the first method, we use the network to reconstruct
directly unlensing $Q$ and $U$ maps from observed $Q$ and $U$ maps. For the
second method, we first use the network to reconstruct lensing B map from
observed $Q$ and $U$ maps, then delens observed BB power spectrum by
subtracting this reconstructed lensing BB power spectrum. Note that our
reconstructed maps for these two methods contain instrumental noise, and we
use the cross-correlation technique to suppress the instrumental noise effect
on the power spectrum.

For the reconstruction of unlensing CMB $Q$ and $U$ maps, from map inspection,
our network shows a good delening performance. The recovery of E-mode power
spectrum is very consistent with the primordial EE spectrum across the entire
range of angular scale we consider. The recovery of the B-mode power spectrum
at noise levels of 0, 1, 2 $\mu$K-arcmin is greater than about 98% at the
angular scales of $\ell<150$ but gradually deviates from the primordial BB as
$\ell$ increases.

For reconstruction of lensing B map, our network can cleanly reconstruct
lensing B map. The recovery of the lensing B-mode power spectrum is greater
than about 99% at the angular scales of $\ell>200$ at noise levels of 0, 1, 2
$\mu$K-arcmin, slightly deviates from the truth lensing BB at the angular
scales of $\ell<200$. We delens observed B-mode power spectrum by subtracting
reconstructed lensing B-mode spectrum. The results show that the recovery of
the tensor BB spectrum at noise levels of 0, 1, 2 $\mu$K-arcmin is greater
than about 98% at the scales of $\ell<120$. At about $\ell=160$, the recovery
of tensor BB power spectrum is around 71%.

We will further investigate the ability of our method to other aspects, such
as the reconstruction of foregrounds and the component separation of future
radio surveys, in future works

## Acknowledgements

J.-Q. Xia is supported by the National Science Foundation of China under
grants No. U1931202 and 12021003; the National Key R&D Program of China No.
2017YFA0402600 and 2020YFC2201603.

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