if $y,y^{\prime}\in M$ and $d_{g}(y,x)=d_{g}(y^{\prime},x)$ for all $x$ in
some open set ${\mathcal{U}}\subset M$, then $y=y^{\prime}$. ∎

We now proceed to prove Proposition 4.7. We first make some reductions using
the continuity of the distance function. By Proposition 4.2, for both $j=1,2$,
$\gamma_{g_{j}}([0,R_{0}],\xi^{\prime}_{0})\subset M_{j}$ are minimizing
segments between ${\bf 0}$ and $\gamma_{g_{j}}(R_{0},\xi^{\prime}_{0})\in
M_{j}$. Taking $R_{0}\in(0,T)$ slightly smaller we may assume without loss of
generality that both

$\displaystyle\gamma_{g_{1}}([0,R_{0}],\xi^{\prime}_{0}),\
\gamma_{g_{2}}([0,R_{0}],\xi^{\prime}_{0})\ {\mbox{are the unique minimizers
and have no conjugate points.}}$ (4.44)

Another observation is that with $y_{1}\in M_{1}$ and $y_{2}\in M_{2}$ fixed,
it suffices to prove (4.43) for a dense subset of $x\in{\mathcal{U}}$. As such
we only need to prove (4.43) for $x\in{\mathcal{U}}$ which is joined to
$y_{1}\in M_{1}$ and $y_{2}\in M_{2}$ by unique minimizing geodesics in
$M_{1}$ and $M_{2}$ which do not contain conjugate points. Furthermore we
assume that

$\displaystyle
x\notin\gamma_{g_{1}}([-\delta_{0},\delta_{0}],\xi^{\prime}_{0})=\gamma_{g_{2}}([-\delta_{0},\delta_{0}],\xi^{\prime}_{0}).$
(4.45)

To this end let $x_{1}\in{\mathcal{U}}$ satisfy the above assumptions. Let
$\gamma_{g_{2}}([0,R_{1}],\xi^{\prime}_{1})$ be the unique minimizing segment
between $x_{1}$ and $y_{2}$ in $M_{2}$ that does not contain conjugate points.
The distance $d_{g_{2}}(x_{1},y_{2})$ is given by

$\displaystyle R_{1}=d_{g_{2}}(x_{1},y_{2}).$ (4.46)

Again, by density, we may without loss of generality assume that

$\displaystyle\dot{\gamma}_{g_{2}}(R_{0},\xi^{\prime}_{0})\neq-\dot{\gamma}_{g_{2}}(R_{1},\xi^{\prime}_{1}).$
(4.47)

By the fact that $\gamma_{g_{2}}([0,R_{0}],\xi^{\prime}_{0})$ is the unique
minimizer between end points, (4.47) implies

$\displaystyle{\bf
0}\notin\gamma_{g_{2}}((0,2R_{0}],\xi^{\prime}_{0})\cup\gamma_{g_{2}}([0,R_{0}+R_{1}],\xi^{\prime}_{1}).$
(4.48)

Note that since both geodesic segments
$\gamma_{g_{2}}([0,R_{0}],\xi^{\prime}_{0})$ and
$\gamma_{g_{2}}([0,R_{1}],\xi^{\prime}_{1})$ are unique minimizers, the two
geodesic segments only intersect at the point $y_{2}$.

We set
$\eta^{\prime}_{l}:=-\dot{\gamma}_{g_{2}}(R_{l},\xi^{\prime}_{l})^{\flat}$ for
$l=0,1$ and use Lemma 2.7 to choose $\eta^{\prime}_{2}\in
S^{*}_{y_{2}}M_{2}\cap\operatorname{span}\\{\eta_{0}^{\prime},\eta_{1}^{\prime}\\}$
arbitrarily close to $\eta^{\prime}_{0}$ so that

$\displaystyle{\rm
Dim}\left(\operatorname{span}\\{-dt-\eta^{\prime}_{0},-dt-\eta^{\prime}_{1},-dt-\eta^{\prime}_{2}\\}\right)=3,\
{\rm
Dim}(\operatorname{span}\\{\eta_{0}^{\prime},\eta_{1}^{\prime},\eta_{2}^{\prime}\\})=2.$
(4.49)

Set $R_{2}:=R_{0}$. By Lemma 2.1 if $\eta^{\prime}_{2}$ is chosen close to
$\eta_{0}^{\prime}$, we can conclude that
$\gamma_{g_{2}}([0,R_{2}],\eta^{\prime}_{2})$ is a unique minimizing segment
containing no conjugate points. Set

$\xi^{\prime}_{2}:=-\dot{\gamma}_{g_{2}}(R_{2},\eta^{\prime}_{2})^{\flat}$

so that

$y_{2}=\gamma_{g_{2}}(R_{0},\xi^{\prime}_{2})=\gamma_{g_{2}}(R_{0},\xi^{\prime}_{0})=\gamma_{g_{2}}(R_{1},\xi^{\prime}_{1}).$

If $\eta^{\prime}_{2}$ is chosen sufficiently close to $\eta^{\prime}_{0}$,
condition (4.47) allows us to assert that

$\displaystyle\dot{\gamma}_{g_{2}}(R_{l},\xi^{\prime}_{0})\neq-\dot{\gamma}_{g_{2}}(R_{k},\xi^{\prime}_{1})$
(4.50)

for $l,k\in\\{0,1,2\\}$.

Due to (4.48), if $\eta^{\prime}_{2}$ is chosen sufficiently close to
$\eta^{\prime}_{0}$ we have

$\displaystyle{\bf
0}\notin\gamma_{g_{2}}((0,2R_{0}],\xi^{\prime}_{0})\cup\gamma_{g_{2}}([0,R_{0}+R_{1}],\xi^{\prime}_{1})\cup\gamma_{g_{2}}([0,2R_{0}],\xi^{\prime}_{2}).$
(4.51)

By Proposition 4.1 the same holds for the $g_{1}$ geodesics:

$\displaystyle{\bf
0}\notin\gamma_{g_{1}}((0,2R_{0}],\xi^{\prime}_{0})\cup\gamma_{g_{1}}([0,R_{0}+R_{1}],\xi^{\prime}_{1})\cup\gamma_{g_{1}}([0,2R_{0}],\xi^{\prime}_{2}).$
(4.52)

Set $t_{0}=t_{2}=0$ and $t_{1}=R_{0}-R_{1}$ so that $t_{l}+R_{l}=R_{0}$ for
all $l=0,1,2$. Note that by (4.46) this means

$\displaystyle d_{g_{2}}(x_{1},y_{2})=R_{0}-t_{1}.$ (4.53)

Also, by the strict triangle inequality, $d_{g_{2}}(x_{0},x_{1})>|t_{1}|$ and
$d_{g_{2}}(x_{2},x_{1})>|t_{1}|$. So

$\displaystyle(t_{l},x_{l})\notin I_{g_{2}}^{+}(t_{k},x_{k}),\ k\neq l.$
(4.54)

Define the lightlike covectors

$\displaystyle\xi_{l}=-dt+\xi^{\prime}_{l}\in
T^{*}_{(t_{l},x_{l})}{\mathcal{M}}_{2}$ (4.55)

for $l=0,1,2$ so that

$\displaystyle(R_{0},y_{2})\in\bigcap_{l=0}^{2}\pi\circ{\rm
FLO}_{g_{2}}^{+}(t_{l},x_{l},\xi_{l})$ (4.56)

By (4.52) there exists an $h_{0}>0$ and $\delta>0$ such that

$\displaystyle B_{G}(2R_{0},{\bf
0};\delta)\cap\left(\bigcup_{l=0}^{2}\pi\circ{\rm FLO}_{g_{1}}^{+}({\rm
ccl}\mathcal{B}_{h_{0}}(\xi_{l})\cap{\rm Char}_{g_{1}})\right)=\emptyset$
(4.57)

Furthermore we can choose the $\delta>0$ in (4.57) to satisfy

######  Lemma 4.9.

i)There exists a $\delta>0$ and $h_{0}>0$ such that if
$\xi\in\mathcal{B}_{h_{0}}(\xi_{0})\cap{\rm Char}_{g_{1}}$ then the lightlike
geodesic segment

$I_{g_{1}}^{-}(B_{G}(2R_{0},{\bf 0};\delta))\cap\pi\circ{\rm
FLO}_{g_{1}}^{+}(0,{\bf 0},\xi)$

is the unique lightlike geodesic segment joining any two points on it.
ii) $I_{g_{1}}^{-}(2R_{0},{\bf 0})\cap\pi\circ{\rm FLO}_{g_{1}}^{+}(0,{\bf
0},\xi_{0})\subset[0,R_{0}]\times M_{1}$.

###### Proof.

i) By assumption $\gamma_{g_{1}}([0,R_{0}];\xi^{\prime}_{0})$ is the unique
minimizer between end points and does not have conjugate points. By Lemma 2.1
there exists an open set $U^{\prime}\subset S^{*}_{\bf 0}{\mathcal{U}}$ and
$\delta^{\prime}>0$ such that if $\xi^{\prime}\in U^{\prime}$ then
$\gamma_{g_{1}}([0,R_{0}+\delta^{\prime}],\xi^{\prime})$ is the unique
minimizer between end points without conjugate points. A consequence of this
is that

$\displaystyle
d_{g_{1}}\left(\gamma_{g_{1}}(R_{0}+\delta^{\prime}/2+t,\xi^{\prime}),{\bf
0}\right)>R_{0}+\delta^{\prime}/2-t$ (4.58)

for all $t\geq 0$ and $\xi^{\prime}\in U^{\prime}$. Furthermore for all
$\xi^{\prime}\in U^{\prime}$, the lightlike geodesic segment

$\\{\pi\circ e_{+}(s,0,{\bf 0},-dt+\xi^{\prime})\mid
s\in[0,R_{0}+\delta^{\prime}]\\}$

is the unique lightlike geodesic joining any two points on it. Due to (4.58),
for $\xi^{\prime}\in U^{\prime}$

$I^{-}_{g_{1}}(2R_{0}+\delta^{\prime}/2,{\bf 0})\cap{\rm
FLO}_{g_{1}}^{+}(0,{\bf
0},-dt+\xi^{\prime})\subset\\{(t,x)\in{\mathcal{M}}_{1}\mid
t\in(0,R_{0}+\delta^{\prime}/2)\\}.$

So if we choose $h_{0}>0$ to satisfy

$\left(\mathcal{B}_{h_{0}}(\xi_{0})\cap{\rm
Char}_{g_{1}}\right)/\mathbb{R}^{+}\subset\\{-dt+\xi^{\prime}\mid\xi^{\prime}\in
U^{\prime}\\},$

then for any $\xi\in\mathcal{B}_{h_{0}}(\xi_{0})\cap{\rm Char}_{g_{1}}$, the
geodesic segment

$I_{g_{1}}^{-}(2R_{0}+\delta^{\prime}/2,{\bf 0})\cap\pi\circ{\rm
FLO}_{g_{1}}^{+}(0,{\bf 0},\xi)$

is the only lightlike geodesic segment joining any two points on it.
Therefore, the lemma is verified if $\delta>0$ is chosen small enough so that
$B_{G}(2R_{0},{\bf 0};\delta)\subset\subset
I_{g_{1}}^{-}(2R_{0}+\delta^{\prime}/2,{\bf 0})$.

ii) This can be seen by taking $\xi^{\prime}=\xi^{\prime}_{0}$ in (4.58) and
observing that

$I_{g_{1}}^{-}(2R_{0},{\bf 0})=\\{(2R_{0}-t,x)\in{\mathcal{M}}_{1}\mid
d_{g_{1}}(x,{\bf 0})\leq t,\ t\geq 0\\}.$

∎

The conditions (4.49), (4.56), (4.54),(4.50) allows us to evoke Proposition
3.7 to conclude

######  Lemma 4.10.

We have that

  1. (1)

For each $l=0,1,2$ there exists a sequence $\xi_{l;j}\in
T^{*}_{(t_{l},x_{l})}{\mathcal{M}}_{2}\cap{\rm Char}_{g_{2}}$ converging to
$\xi_{l}$.

  2. (2)

For each $l=0,1,2$ and $j\in\mathbb{N}$ large there is an $h_{j}\in(0,h_{0})$
with

$\mathcal{B}_{h_{j}}(\xi_{l;j})\subset\subset\mathcal{B}_{h_{0}}(\xi_{l})$

so that if $h\in(0,h_{j})$ then there exists sources $f_{l;j}(\cdot;h)$ of the
form (2.12) such that

$\xi_{l;j}\in\operatorname{WF}(f_{l;j}(\cdot;h))\subset{\rm
ccl}(\mathcal{B}_{h}(\xi_{l;j})\cup\mathcal{B}_{h}(-\xi_{l;j})),\
\sigma(f_{l;j}(\cdot;h))(t_{l},x_{l},\xi_{l;j})\neq 0.$

  3. (3)

A sequence $(\tilde{T}_{j},\tilde{x}_{j})\to(2R_{0},{\bf 0})$,
$\tilde{T}_{j}>2R_{0}$, and $\tilde{x}_{j}\neq{\bf 0}$.

  4. (4)

For any $j\in\mathbb{N}$ large and $h\in(0,h_{j})$ as above, and any
$a\in(0,h)$ sufficiently small, we have sources $f_{3;j}(\cdot;a)$ and
$f_{4;j}(\cdot;a)$ of the form (3.55) which are supported in
$B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)$.

If $(t,x)\mapsto v^{j}(t,x;h,a)$ are solutions of (4.1) in ${\mathcal{M}}_{2}$
with source

$f^{j}(\cdot;h,a):=\sum_{l=0}^{2}\epsilon_{l}f_{l;j}(\cdot;h)+\sum_{l=3}^{4}\epsilon_{l}f_{l;j}(\cdot;a)$

then for each $j\in\mathbb{N}$ large

$\displaystyle\tilde{T}_{j}+d_{g_{2}}(\tilde{x}_{j},{\bf 0})\in{\rm
singsupp}(v^{j}_{01234}(\cdot,{\bf 0};h,a)).$ (4.59)

for all $0<a<h<h_{j}$ small.

Note that we now explicitly write out the dependence of the sources $f_{l;j}$
and solution $v^{j}$ on the parameters $0<a<h<h_{j}$.

Let $u^{j}(\cdot;h,a)$ solve (4.1) with metric $g_{1}$ and the same source as
$v^{j}(\cdot;h,a)$. The fivefold interaction of the nonlinear wave
$u^{j}(\cdot;h,a)$ is

$\displaystyle\Box_{g_{1}}u^{j}_{01234}(\cdot;h,a)=\sum_{\sigma\in
S_{5}}u^{j}_{\sigma(0)\sigma(1)\sigma(2)}(\cdot;h,a)u^{j}_{\sigma(3)}(\cdot;h,a)u^{j}_{\sigma(4)}(\cdot;h,a),\
u^{j}_{01234}(\cdot;h,a)\mid_{t<-1}=0$

We write the solution $u^{j}_{01234}(\cdot;h,a)$ of (4.3) as $u^{j}_{\rm
reg}+u^{j}_{\rm sing}(\cdot;h,a)$ where

$\displaystyle\Box_{g_{1}}u^{j}_{\rm reg}=\sum_{\sigma\in S_{5}\backslash
S_{3}}u^{j}_{\sigma(0)\sigma(1)\sigma(2)}(\cdot;h,a)u^{j}_{\sigma(3)}(\cdot;h,a)u^{j}_{\sigma(4)}(\cdot;h,a),\
u^{j}_{\rm reg}\mid_{t<-1}=0$ (4.61)

and

$\displaystyle\Box_{g_{1}}u^{j}_{\rm
sing}(\cdot;h,a)=6u^{j}_{012}(\cdot;h,a)u^{j}_{3}(\cdot;h,a)u_{4}^{j}(\cdot;h,a),\
u^{j}_{\rm sing}(\cdot;h,a)\mid_{t<-1}=0.$ (4.62)

Here we denote $S_{3}\subset S_{5}$ as elements of the permutation group which
maps the set of three letters $\\{0,1,2\\}$ to itself. Repeating the argument
of Lemma 3.8 we get that

######  Lemma 4.11.

If $j\in\mathbb{N}$ is chosen large enough and $a\in(0,h)$ is chosen small
enough so that $B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)\subset\subset
B_{G}(2R_{0},{\bf 0};\delta)$ and
$B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)\cap(\mathbb{R}\times\\{{\bf
0}\\})=\emptyset$, then

$(\tilde{T}_{j}+d_{g}(\tilde{x}_{j},{\bf 0}),{\bf 0})\notin{\rm
singsupp}(u_{\rm reg}).$

###### Proof.

This is similar to proof of Lemma 3.8 so we will only give a brief sketch. To
simplify notation we define

$f_{\rm source}:=\sum_{\sigma\in S_{5}\backslash
S_{3}}u^{j}_{\sigma(0)\sigma(1)\sigma(2)}(\cdot;h,a)u^{j}_{\sigma(3)}(\cdot;h,a)u^{j}_{\sigma(4)}(\cdot;h,a).$

Note that by (3.57)

$u^{j}_{4}=\chi_{a;j}\in
C^{\infty}_{c}(B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)),\
u^{j}_{3}=\chi_{a;j}\langle D\rangle^{-N}\delta_{{\bf T}_{j}}\in
I({\mathcal{M}},N^{*}{\bf T}_{j}).$

Meanwhile by Proposition 2.9, for $l=0,1,2$

$u^{j}_{l}\in I({\mathcal{M}}_{1},T^{*}_{(t_{l},x_{l})}{\mathcal{M}}_{1},{\rm
FLO}_{g_{1}}^{+}({\rm
ccl}(\mathcal{B}_{h_{0}}(\xi_{l})\cup\mathcal{B}_{h_{0}}(-\xi_{l}))\cap{\rm
Char}_{g_{1}}).$

This combined with flowout condition (4.57) ensures that for $l=0,1,2$,

${\rm
singsupp}(u^{j}_{l})\cap\left(\operatorname{supp}(u_{3}^{j})\cup\operatorname{supp}(u_{4}^{j})\right)=\emptyset.$

Using these facts we can proceed as in Lemma 3.8 to conclude that

${\rm FLO}_{g_{1}}^{-}(\tilde{T}_{j}+d_{g}(\tilde{x}_{j},{\bf 0}),{\bf
0},\xi)\cap\operatorname{WF}(f_{\rm source})=\emptyset$

for all $\xi\in T^{*}_{(\tilde{T}_{j}+d_{g}(\tilde{x}_{j},{\bf 0}),{\bf
0})}{\mathcal{U}}\cap{\rm Char}_{g_{1}}$. The lemma then follows from Thm
23.2.9 of [H0̈7]. ∎

With these facts we can now give the

###### Proof of Proposition 4.7.

From (4.59) we can use Lemma 2.16 to deduce

$\tilde{T}_{j}+d_{g_{2}}(\tilde{x}_{j},{\bf 0})\in{\rm
singsupp}\left(\partial^{5}_{\epsilon_{0}\dots\epsilon_{4}}\left(v^{j}(\cdot,{\bf
0};h,a)\right)\mid_{\epsilon_{0}=\cdots=\epsilon_{4}=0}\right).$

By the fact that the source-to-solution maps agree, $t\mapsto u^{j}(t,{\bf
0};h,a)$ is the same as $t\mapsto v^{j}(t,{\bf 0};h,a)$. So we have that

$\tilde{T}_{j}+d_{g_{2}}(\tilde{x}_{j},{\bf 0})\in{\rm
singsupp}\left(\partial^{5}_{\epsilon_{0}\dots\epsilon_{4}}\left(u^{j}(\cdot,{\bf
0};h,a)\right)\mid_{\epsilon_{0}=\cdots=\epsilon_{4}=0}\right).$

Lemma 2.16 then implies $\tilde{T}_{j}+d_{g}(\tilde{x}_{j},{\bf 0})\in{\rm
singsupp}(u^{j}_{01234}(\cdot,{\bf 0};h,a))$. By Lemma 4.11 we have that

$\displaystyle(\tilde{T}_{j}+d_{g}(\tilde{x}_{j},{\bf 0}),{\bf 0})\in{\rm
singsupp}(u^{j}_{\rm sing}(\cdot;h,a)).$ (4.63)

Note that for all $a>0$ small, the distribution $u^{j}_{\rm sing}(\cdot;h,a)$
solves (4.62) and the source term of (4.62) is supported in
$B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)$ since
$\operatorname{supp}(u_{4}^{j}(\cdot;a))\subset
B_{G}(\tilde{T}_{j},\tilde{x}_{j};a)$ by (3.57). Note that by Condition (3) of
Lemma 4.10, $x_{j}\neq{\bf 0}$. The solution $u^{j}_{4}(\cdot;a)$ is smooth
and the wavefront set of $u^{j}_{3}(\cdot;a)$ is spacelike. So in order for
(4.63) to hold,

$B_{G}((\tilde{T}_{j},\tilde{x}_{j}),a)\cap{\rm
singsupp}(u^{j}_{012}(\cdot;h))\neq\emptyset$

for all $a>0$. We remark that $u^{j}_{012}(\cdot;h)$ does not depend on the
parameter $a>0$ since $f_{l;j}(\cdot;h)$ depends only on the parameter $h>0$
for $l=0,1,2$. So taking intersection over all $a>0$, we conclude that, for
each fixed $j\in\mathbb{N}$ large, if $h\in(0,h_{j})$ is sufficiently small,

$\displaystyle(\tilde{T}_{j},\tilde{x}_{j})\in{\rm
singsupp}(u^{j}_{012}(\cdot;h)).$ (4.64)

Fix $j\in\mathbb{N}$ large and for each $h>0$ sufficiently small
$u^{j}_{012}(\cdot;h)$ solves

$\Box_{g_{0}}u^{j}_{012}(\cdot;h)=-6u^{j}_{0}(\cdot;h)u^{j}_{1}(\cdot;h)u^{j}_{2}(\cdot;h),\
u^{j}_{012}(\cdot;h)\mid_{t<-1}=0$

with each of $u^{j}_{l}(\cdot;h)$ solving linear inhomogeneous wave equation
with source $f_{l;j}(\cdot;h)$ so that ${\rm
singsupp}(u^{j}_{l}(\cdot;h))\subset\pi\circ{\rm
FLO}_{g_{1}}^{+}(\operatorname{WF}(f_{l;j}(\cdot;h))\cap{\rm Char}_{g_{1}})$
by Proposition 2.9.

Take $j\in\mathbb{N}$ large enough so that $(\tilde{T}_{j},\tilde{x}_{j})\in
B_{G}((2R_{0},{\bf 0}),\delta)$ where $\delta>0$ is chosen so that (4.57) and
the conclusion of Lemma 4.9 holds. By Lemma 2.11 and (4.57), in order for
(4.64) to hold we must have

$\displaystyle
I_{g_{1}}^{-}((\tilde{T}_{j},\tilde{x}_{j}))\cap\bigcap_{l=0}^{2}\pi\circ{\rm
FLO}_{g_{1}}^{+}(\operatorname{WF}(f_{l;j}(\cdot;h))\cap{\rm
Char}_{g_{1}})\neq\emptyset$ (4.65)

for all $h\in(0,h_{j})$.

By Condition (2) of Lemma 4.10,
$\operatorname{WF}(f_{l;j}(\cdot;h))\subset{\rm
ccl}\mathcal{B}_{h}(\pm\xi_{l;j})$ for $l=0,1,2$ so (4.65) becomes

$\displaystyle
I_{g_{1}}^{-}((\tilde{T}_{j},\tilde{x}_{j}))\cap\bigcap_{l=0}^{2}\pi\circ{\rm
FLO}_{g_{1}}^{+}({\rm ccl}\mathcal{B}_{h}(\xi_{l;j})\cap{\rm
Char}_{g_{1}})\neq\emptyset$ (4.66)

for all $h>0$. Furthermore we must have that

$\displaystyle\forall h\in(0,h_{j}),\exists\eta_{h}\in
T_{(t_{h},x_{h})}^{*}{\mathcal{M}}_{1}/\mathbb{R}^{+}\cap{\rm Char}_{g_{1}},\
(t_{h},x_{h})\in\bigcap_{l=0}^{2}\pi\circ{\rm FLO}_{g_{1}}^{+}({\rm
ccl}\mathcal{B}_{h}(\xi_{l;j})\cap{\rm Char}_{g_{1}})$ $\displaystyle{\rm
s.t.}\ (\tilde{T}_{j},\tilde{x}_{j})\in\pi\circ{\rm
FLO}^{+}_{g_{1}}(t_{h},x_{h},\eta_{h})$ (4.67)

for otherwise $(\tilde{T}_{j},\tilde{x}_{j})\notin{\rm
singsupp}(u_{012}^{j}(\cdot;h))$ by Thm 23.2.9 of [H0̈7].

Since this holds for all $h>0$, compactness then dictates that there exists
$(\hat{R}_{j},\hat{y}_{j})\in{\mathcal{M}}_{1}$ such that

$\displaystyle(\hat{R}_{j},\hat{y}_{j})\in
I_{g_{1}}^{-}((\tilde{T}_{j},\tilde{x}_{j}))\cap\bigcap_{l=0}^{2}\pi\circ{\rm
FLO}_{g_{1}}^{+}(t_{l},x_{l},\xi_{l;j}).$ (4.68)

By Lemma 4.9, $(\hat{R}_{j},\hat{y}_{j})$ is unique element in (4.68).
Therefore we have that

$\displaystyle(\hat{R}_{j},\hat{y}_{j})=I_{g_{1}}^{-}((\tilde{T}_{j},\tilde{x}_{j}))\cap\bigcap_{l=0}^{2}\pi\circ{\rm
FLO}_{g_{1}}^{+}(t_{l},x_{l},\xi_{l;j}).$ (4.69)

And by (4.3), there is a covector $\hat{\eta}_{j}\in
T^{*}_{(\hat{R}_{j},\hat{y}_{j})}{\mathcal{M}}_{1}/\mathbb{R}^{+}\cap{\rm
Char}_{g_{1}}$ such that

$\displaystyle(\tilde{T}_{j},\tilde{x}_{j})\in\pi\circ{\rm
FLO}_{g_{1}}^{+}(\hat{R}_{j},\hat{x}_{j},\hat{\eta}_{j}).$ (4.70)

Taking a converging subsequence as $j\to\infty$ in (4.69) and (4.70) we have,
by Conditions (1) and (3) of Lemma 4.10,

$\displaystyle(\hat{R}_{0},\hat{y}_{0})=I_{g_{1}}^{-}(2R_{0},{\bf
0})\cap\bigcap_{l=0}^{2}{\rm FLO}_{g_{1}}^{+}(t_{l},x_{l},\xi_{l}),$ (4.71)
$\displaystyle(2R_{0},{\bf 0})\in\pi\circ{\rm
FLO}_{g_{1}}^{+}(\hat{R}_{0},\hat{y}_{0},\hat{\eta})$ (4.72)

for some $(\hat{R}_{0},\hat{y}_{0})\in{\mathcal{M}}_{1}$ and $\hat{\eta}\in
T^{*}_{(\hat{R}_{0},\hat{y}_{0})}{\mathcal{M}}_{1}\cap{\rm Char}_{g_{1}}$.
Since $\xi_{0}\in T^{*}_{(0,{\bf 0})}{\mathcal{M}}_{1}\cap{\rm Char}_{g_{1}}$
(see (4.55)), this says that

$\displaystyle(\hat{R}_{0},\hat{y}_{0})\in{\rm FLO}_{g_{1}}^{+}(0,{\bf
0},\xi_{0})$ (4.73)

which in turn implies that

$\displaystyle\hat{y}_{0}=\gamma_{g_{1}}(\hat{R}_{0},\xi^{\prime}_{0}).$
(4.74)

Evoke part ii) of Lemma 4.9 combined with (4.73) we conclude

$\displaystyle\hat{R}_{0}\leq R_{0}.$ (4.75)

By (4.44), the geodesic segment $\gamma_{g_{1}}([0,R_{0}],\xi^{\prime}_{0})$
is the unique minimizer between the end points ${\bf 0}$ and $y_{1}$. So
(4.75) and (4.74) combined to give

$\displaystyle d_{g_{1}}(\hat{y}_{0},y_{1})=R_{0}-\hat{R}_{0}.$ (4.76)

By (4.55), $\xi_{1}\in T^{*}_{(t_{1},x_{1})}{\mathcal{M}}_{1}\cap{\rm
Char}_{g_{1}}$. So since $(\hat{R}_{0},\hat{y}_{0})\in{\rm
FLO}_{g_{1}}^{+}(t_{1},x_{1},\xi_{1})$ by (4.71), we can conclude that

$\displaystyle\hat{y}_{0}=\gamma_{g_{1}}(\hat{R}_{0}-t_{1},\xi_{1}^{\prime}),\
x_{1}=\gamma_{g_{1}}(0,\xi^{\prime}_{1}).$ (4.77)

Combining (4.77) and (4.76) we get

$d_{g_{1}}(x_{1},y_{1})\leq
d_{g_{1}}(x_{1},\hat{y}_{0})+d_{g_{1}}(\hat{y}_{0},y_{1})\leq(\hat{R}_{0}-t_{1})+(R_{0}-\hat{R}_{0})=R_{0}-t_{1}.$

So by (4.53) we get that

$d_{g_{1}}(x_{1},y_{1})\leq d_{g_{2}}(x_{1},y_{2}).$

∎

###  4.4. Recover Riemannian Structure from Distance Data

We now complete the proof of Theorem 1.1 with the following proposition. Let
$(M_{i},g_{i})$, $i=1,2$, be complete Riemannian manifolds. For some $x_{i}\in
M_{i}$ and $R>0$, we define

$\displaystyle V_{i}:=\big{\\{}\xi^{\prime}\in T_{x_{i}}M_{i}\ \big{|}\
d_{g_{i}}(x_{i},\exp_{x_{i}}(\xi^{\prime}))=\|\xi^{\prime}\|_{g_{i}}<T\big{\\}}.$

Notice that $V_{i}$ is precisely the set of those tangent vectors
$\xi^{\prime}\in T_{x_{i}}M_{i}$ whose associated geodesic
$\gamma(\cdot,\xi^{\prime}):[0,1]\to M_{i}$,
$\gamma(t,\xi^{\prime})=\exp_{x_{i}}(t\xi^{\prime})$ is length-minimizing.

######  Proposition 4.12.

Assume that, for some open neighborhoods ${\mathcal{U}}_{i}\subset
B_{g_{i}}(x_{i},T)$ of $x_{i}$, there exists an isometry
$\psi:({\mathcal{U}}_{1},g_{1})\to({\mathcal{U}}_{2},g_{2})$ such that

  * (i)

$d\psi(x_{1})V_{1}=V_{2}$,

  * (ii)

$d_{g_{1}}(y,\exp_{x_{1}}(\xi^{\prime}))=d_{g_{2}}(\psi(y),\exp_{x_{2}}(d\psi(x_{1})\xi^{\prime}))$
for all $y\in U_{1}$ and $\xi^{\prime}\in V_{1}$.

Then, there is an extension of $\psi$ to an isometry

$\psi:(B_{g_{1}}(x_{1},T),g_{1})\to(B_{g_{2}}(x_{2},T),g_{2}).$

###### Proof.

Since the Riemannian manifolds $(M_{i},g_{i})$ are complete, every point of
$B_{g_{i}}(x_{i},T)$ can be connected to $x_{i}$ by means of a length-
minimizing geodesic. Namely,

$\exp_{x_{i}}(V_{i})=B_{g_{i}}(x_{i},T).$

We claim that there exists a continuous bijection $\phi:B_{g_{1}}(x_{1},T)\to
B_{g_{2}}(x_{2},T)$ such that

$\displaystyle\phi(\exp_{x_{1}}(\xi^{\prime}))=\exp_{x_{2}}(d\psi(x_{1})\xi^{\prime}),\qquad\forall\xi^{\prime}\in
V_{1}$

Indeed, since the isometry $\psi$ maps geodesics to geodesics, such a $\phi$
is clearly well defined on a Riemannian ball
$B_{g_{1}}(x_{1},\epsilon)\subset{\mathcal{U}}_{1}$, and indeed
$\phi|_{B_{g_{1}}(x_{1},\epsilon)}=\psi|_{B_{g_{1}}(x_{1},\epsilon)}$. Assume
that there exist two distinct vectors $\xi^{\prime},\eta^{\prime}\in V_{1}$
such that $y_{1}:=\exp_{x_{1}}(\xi^{\prime})=\exp_{x_{1}}(\eta^{\prime})$. All
we need to show in order to have a well defined continuous bijection $\phi$ is
that

$\displaystyle\exp_{x_{2}}(d\psi(x_{1})\xi^{\prime})=\exp_{x_{2}}(d\psi(x_{1})\eta^{\prime}).$
(4.78)

We set $z_{1}:=\exp_{x_{1}}(\delta\eta^{\prime})\in U_{1}$ for some $\delta>0$
small enough, $z_{2}:=\exp_{x_{2}}(d\psi(x_{1})\delta\eta^{\prime})$, and
$y_{2}:=\exp_{x_{2}}(d\psi(x_{1})\xi^{\prime})$. We have

$\displaystyle d_{g_{2}}(x_{2},y_{2})$
$\displaystyle=\|\xi^{\prime}\|_{g_{1}}=d_{g_{1}}(x_{1},y_{1})=d_{g_{1}}(x_{1},z_{1})+d_{g_{1}}(z_{1},y_{1})=d_{g_{2}}(x_{2},z_{2})+d_{g_{2}}(z_{2},y_{2}),$

where the last equality follows by assumption (ii). This implies that $z_{2}$
lies on a length-minimizing geodesic segment joining $x_{2}$ and $y_{2}$, and
therefore (4.78) holds.

We now prove that $\phi$ is a diffeomorphism. Fix an arbitrary point $y_{1}\in
B_{g_{1}}(x_{1},T)$, and consider the unit-speed length-minimizing geodesic
$\gamma_{1}:[0,\ell]\to M_{1}$ joining $x_{1}=\gamma_{1}(0)$ and
$y_{1}=\gamma_{1}(\ell)$. We fix $\delta>0$ small enough so that
$z_{1}:=\gamma_{1}(\delta)\in B_{g_{1}}(x_{1},\epsilon)$. Notice that
$\gamma_{1}|_{[\delta,\ell]}$ is the unique length-minimizing geodesic segment
joining $z_{1}$ and $y_{1}$, and does not contain conjugate points.
Analogously, $\gamma_{2}:=\phi\circ\gamma_{1}|_{[\delta,\ell]}$ is the unique
length-minimizing geodesic segment joining $z_{2}:=\phi(z_{1})$ and
$y_{2}:=\phi(y_{1})$, and does not have conjugate points. Therefore, the
Riemannian distances $d_{g_{i}}$ are smooth in an open neighborhood
$Z_{i}\times Y_{i}$ of $(z_{i},y_{i})$. We take $Z_{i}$ to be small enough so
that $Z_{1}\subset B_{g_{1}}(x_{1},\epsilon)$ and $\phi(Z_{1})=Z_{2}$. For
each $w\in Z_{i}$, the derivative $\partial_{y_{i}}d_{g_{i}}(w,y_{i})\in
S_{y_{i}}^{*}M$ is precisely the $g_{i}$-dual of the tangent vector to the
unique length-minimizing geodesic segment joining $w$ and $y_{i}$. Therefore,
for a generic triple of points $w_{1},w_{2},w_{3}\in Z_{1}$ sufficiently close
to $z_{1}$, the triple

$\partial_{y_{1}}d_{g_{1}}(w_{1},y_{1}),\partial_{y_{1}}d_{g_{1}}(w_{2},y_{1}),\partial_{y_{1}}d_{g_{1}}(w_{3},y_{1})$

is a basis of $T_{y_{1}}^{*}M_{1}$, and the triple

$\partial_{y_{2}}d_{g_{2}}(\phi(w_{1}),y_{2}),\partial_{y_{2}}d_{g_{2}}(\phi(w_{2}),y_{2}),\partial_{y_{2}}d_{g_{2}}(\phi(w_{3}),y_{2})$

is a basis of $T_{y_{2}}^{*}M_{2}$. This implies that, up to shrinking the
open neighborhoods $Y_{i}$ of $y_{i}$, the maps

$\displaystyle\kappa_{1}:Y_{1}\to\mathbb{R}^{3},\qquad\kappa_{1}(y)$
$\displaystyle=(d_{g_{1}}(w_{1},y),d_{g_{1}}(w_{2},y),d_{g_{1}}(w_{3},y)),$
$\displaystyle\kappa_{2}:Y_{2}\to\mathbb{R}^{3},\qquad\kappa_{2}(y)$
$\displaystyle=(d_{g_{2}}(\phi(w_{1}),y),d_{g_{2}}(\phi(w_{2}),y),d_{g_{2}}(\phi(w_{3}),y))$

are diffeomorphisms onto their images. By assumption (ii), we have
$\kappa_{1}=\kappa_{2}\circ\phi|_{Y_{1}}$, and therefore
$\phi|_{Y_{1}}=\kappa_{2}^{-1}\circ\kappa_{1}$. This proves that $\phi$ is a
local diffeomorphism at $y_{1}$. Moreover, for all $z\in Z_{1}$, by
differentiating the equality
$d_{g_{1}}(z,y_{1})=d_{g_{2}}(\phi(z),\phi(y_{1}))$ with respect of $y_{1}$,
we obtain

$\displaystyle\underbrace{\partial_{y_{1}}d_{g_{1}}(z,y_{1})}_{\in
S^{*}_{y_{1}}M_{1}}=\partial_{y_{2}}d_{g_{2}}(\phi(z),y_{2})d\phi(z_{1})=d\phi(z_{1})^{*}\underbrace{\partial_{y_{2}}d_{g_{2}}(\phi(z),y_{2})}_{\in
S^{*}_{y_{2}}M_{2}}.$

Since this holds for all $z$ in the open set $Z_{1}$, we obtain non-empty open
sets $W_{i}\subset S^{*}_{y_{i}}M_{i}$ such that
$d\phi(y_{1})^{*}W_{2}=W_{1}$. This implies
$d\phi(y_{1})^{*}S^{*}_{y_{2}}M_{2}=S^{*}_{y_{1}}M_{1}$, and therefore
$(\phi^{*}g_{2})|_{y_{1}}=g_{1}|_{y_{1}}$. Since $y_{1}$ is arbitrary, we
conclude that $\phi$ is a Riemannian isometry as claimed. ∎

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the Nash equilibrium i.e., a state beyond which they can not improve their
performance unilaterally [48]. While this cooperative architecture aims to
optimize a global loss function, the optimization problems faced by the
individual networks are fundamentally opposing. Due to this complexity in the
loss function, there can be situations where some minor adjustments in one
network can trigger substantial modifications in the other. Moreover, when
both the networks aim to independently optimize their loss functions without
coordination, attaining the Nash equilibrium can be hard. Such instances of
desynchronization between the networks can lead to instability in the overall
learning process and substantially increase the computation time [221]. To
counter this challenge, recent advancements in GAN architectures have been
focusing on enhancing training stability. The feature matching technique
improves the stability of the GAN framework by introducing an alternative cost
function for $G$ combining the output of the discriminator [202].
Additionally, historical averaging of the parameters [202], unrolled GAN
[219], and gradient penalty [122] strategies mitigate learning instability and
promote convergence of the model.

###  VIII-D Stopping Problem

During GANs training, determining the appropriate time at which the networks
are fully optimized is crucial for addressing the problems related to
overfitting and underfitting. However, in GANs due to the minimax objective
function determining the state of the networks based on their respective loss
functions is impossible. To address this issue related to the GANs stopping
criterion, researchers often employ an early stopping approach where the
training halts based on a predefined threshold or the lack of improvement in
evaluation metrics.

###  VIII-E Internal Distributional Shift

The internal distributional shift often called internal covariate shift refers
to the changing distribution in the network activations of the current layer
w.r.t the previous layer. In the context of GAN, when the generator’s
parameters are updated, the distribution of its output may change, leading to
internal distributional shifts in subsequent layers and causing the
discriminator’s learning to lag behind. This phenomenon affects the
convergence of the GAN training process and the computational complexity of
the network significantly increases to counter the shifts. To address this
issue batch normalization technique is widely adopted in various applications
of GAN [222].

##  IX DISCUSSION

Over the past decade, GANs have emerged as the foremost and pivotal generative
architecture within the areas of computer vision, natural language processing,
and related fields. To enhance the performance of GAN architecture, numerous
studies have focused on the following: (i) the generation of high-quality
samples, (ii) diversity in the simulated samples, and (iii) stabilizing the
training algorithm. Constant efforts and improvements of the GAN model have
resulted in plausible sample generation, text/image-to-image translations,
data augmentation, style transfer, anomaly detection, and other applied
domains.

Recent advancements in machine learning with the help of Diffusion models [22,
223, 224] also known as score-based generative models have made a strong
impression on a variety of tasks including image denoising, image inpainting,
image super-resolution, and image generation. The primary goal of Diffusion
models is to learn the latent structure of the dataset by modeling the way in
which data points diffuse through the latent space. [225] has shown that
Diffusion models outperform GANs on image synthesis due to their better
stability and non-existence of mode collapse. However, the cost of
synthesizing new samples and computational time for making realistic images
lead to its shortcomings when applied to real-time application [226, 227]. Due
to the fact that GANs need fine-tuning in their hyperparameters, Transformers
[19] have been used to enhance the results of GANs that can adopt self-
attention layers. This helps in designing larger models and replacing the
neural network models of $G$ and $D$ within the GAN structure. TransGAN [228]
introduces a GAN architecture without convolutions by using Transformers in
both $G$ and $D$ of the GAN resulting in improved high-resolution image
generation. [229] presented an intersection of GANs and Transformers to
predict pedestrian paths. Although Transformers and their variants have
several advantages, they suffer from high computational (time and resource)
complexity [230]. More recently, physics-informed neural networks (PINN) [20]
was introduced as a universal function approximator that can incorporate
knowledge of physical laws to govern the data in the learning process. PINNs
overcome the low data availability issue [231] in which GANs and Transformers
lack robustness, rendering them ineffective scenarios. A GAN framework based
on a physics-informed (PI) discriminator for uncertainty quantification is
used to inform the knowledge of physics during the learning of both $G$ and
$D$ models. Physics-informed Discriminator GAN (PID-GAN) [232] doesn’t suffer
from an imbalance of generator gradient from multiple losses. Another
architecture namely Physics-informed GAN (PI-GAN) [233] tackles the problem of
sequence generation with limited data. It integrates a transition module in
the generator part that can iteratively construct the sequence with only one
initial point as input. Solving differential equations using GANs to learn the
loss function was presented in the Differential Equation GAN (DEQ-GAN) model
[234]. Combining GANs with PINNs achieved solution accuracies that are
competitive with popularly used numerical methods.

Large language models (LLMs) [21] became a very popular choice for their
ability to understand and generate human language. LLMs are neural networks
that are trained on massive text datasets to understand the relationship
between words and phrases. This enables LLMs to generate text that is both
coherent and grammatically correct. Recently, LLMs and their cousin ChatGPT
revolutionized the field of natural language processing, question-answering,
and creative writing. Additionally, LLMs and their variants are used to create
creative content such as poems, scripts, and codes. GANs and LLMs are two
powerful co-existing models where the former is used to generate realistic
images. Mega-TTS [235] adopt a VQGAN [169] based acoustic model and a latent-
code language model called Prosody-LLM (P-LLM) [236] to solve zero-shot text-
to-speech at scale with intrinsic inductive bias. Future works in the
hybridization of GANs with several other architectures will be a promising
field of future research.

##  X FUTURE RESEARCH DIRECTION

Despite the substantial advancements achieved by GAN-based frameworks over the
past decade, there remain a number of challenges spanning both theoretical and
practical aspects that require further exploration in future research. In this
section, we identify these gaps that necessitate deeper investigation to
enhance our comprehension of GANs. The summary is presented below:

#### Fundamental questions on the theory of GANs

Recent advancements in the theory of GAN by [197, 192, 193] explored the role
of the discriminator family in terms of JS divergence and some large sample
properties (convergence and asymptotic normality) of the parameter describing
the empirically selected generator. However, a fundamental question of how
well GANs can approximate the target distribution $p^{*}$ remained largely
unanswered. From the theoretical perspective, there is still a mystery about
the role and impact of the discriminator on the quality of the approximation.
The universal consistency and the rate of convergence of GANs and their
variants still remain an open problem.

#### Improvement of training stability and diversity

Achieving the Nash equilibrium in GAN frameworks, which is essential for the
generator to learn the actual sample distribution, requires stable training
mechanisms [237, 238]. However, attaining this optimal balance between the
generator and discriminator remains challenging. Various approaches have been
explored, such as WGAN [109], SN-GAN [133], One-sided Label Smoothing [203],
and WGAN with gradient penalty (WGAN-GP) [122], to enhance training stability.
Additionally, addressing mode collapse, a common GAN issue that leads to
limited sample diversity, has prompted strategies like WGAN [109], U-GAN
[219], generator regulating GAN (GRGAN) [239], and Adaptive GAN [240]. Future
research could focus on devising techniques to stabilize GAN training and
alleviate problems like mode collapse through regularization methods,
alternative loss functions, and optimized hyperparameters. Incorporating
methods like multi-modal GANs, designed to generate diverse outputs from a
single input, might contribute to enhancing sample diversity [239].

#### Data scarcity in GAN

Addressing the issue of data scarcity in GANs stands as a crucial research
trajectory. To expand GAN applications, forthcoming investigations could focus
on devising training strategies for scenarios with limited data. Approaches
such as few-shot GANs, transfer learning, and domain adaptation offer the
potential to enhance GAN performance when data is scarce [241, 242]. This
challenge becomes especially pertinent when acquiring substantial datasets
poses difficulties. Additionally, refining training algorithms for maximal
data utility could be pursued. Bolstering GAN effectiveness in low-data
situations holds pivotal significance for broader adoption across various
industries and domains.

#### Ethics and privacy

Since its inception in 2014, GAN development has yielded substantial benefits
in research and real-world applications. However, the inappropriate
utilization of GANs can give rise to latent societal issues such as producing
deceptive content, malicious images, fabricated news, deepfakes, prejudiced
portrayals, and compromising individual safety [243]. To tackle these issues,
the establishment of ethical guidelines and regulations is imperative [244].
Future research avenues might center on developing robust techniques to detect
and alleviate ethical concerns associated with GANs, while also advocating
their ethical and responsible deployment in diverse fields. Essential to this
effort is the creation of forgery detection methods capable of effectively
identifying AI-generated content, including images produced through GANs.
Furthermore, GANs can be susceptible to adversarial attacks, wherein minor
modifications to input data result in visually convincing yet incorrect
outputs [245, 116]. Future investigations could prioritize the development of
robust GANs that can withstand such attacks, alongside methods for identifying
and countering them. Ensuring the integrity and reliability of GANs is of
utmost importance, particularly in contexts like authentication, content
verification, and cybersecurity [246, 216].

#### Real-time implementation and scalability

While GANs have shown immense potential, their resource-intensive nature
hinders real-time usage and scalability. Recent GAN variants like ProGAN [5]
and Att-GAN [148] aim to address this complexity. Future efforts might focus
on crafting efficient GAN architectures capable of generating high-quality
samples in real-time, vital for constrained platforms like mobile devices and
edge computing. Integrating GANs with reinforcement learning, transfer
learning, and supervised learning, as seen in RidgeGAN [10], opens
opportunities for hybrid models with expanded capabilities. Research should
delve into hybrid approaches, leveraging GANs alongside other techniques for
enhanced generative potential. Additionally, exploring multimodal GANs that
produce diverse outputs from multiple modalities can unlock novel avenues for
creating complex data [247].

#### Human-centric GANs

GANs have the potential to enable human-machine creative cooperation [248].
Future research could emphasize human-centric GANs, integrating human
feedback, preferences, and creativity into the generative process. This
direction might pave the way for interactive and co-creative GANs, enabling
the production of outputs aligned with human preferences and needs, while also
involving users in active participation during the generation process.

#### Other innovative applications and industry usage

Initially designed for generating realistic images, GANs have exhibited
impressive performance in computer vision. While their application has
extended to domains like time series generation [103, 102], audio synthesis
[8], and autonomous vehicles [120], their use outside computer vision remains
somewhat constrained. The divergent nature of image and non-image data
introduces challenges, particularly in non-image contexts like NLP, where
discrete values such as words and characters predominate [199]. Future
research can aim to overcome these challenges and enhance GANs’ capabilities
in discrete data scenarios. Furthermore, exploring unique applications of GANs
in fields like finance, education, and entertainment offers the potential to
introduce new possibilities and positively impact various industries [249].
Collaborative efforts across disciplines could also harness diverse expertise,
fostering synergies to enhance GANs’ adaptability across a broad spectrum of
applications [250].

##  XI CONCLUSION

In this article, we presented a GAN survey, GAN variants, and a detailed
analysis of the wide range of GAN applications in several applied domains. In
addition, we reviewed the recent theoretical developments in the GAN
literature and the most common evaluation metrics. Despite all these one of
the core contributions of this survey is to discuss several obstacles of
various GAN architectures and their potential solutions for future research.
Overall, we discuss GANs’ potential to facilitate practical applications not
only in image, audio, and text but also in relatively uncommon areas such as
time series analysis, geospatial data analysis, and imbalanced learning. In
the discussion section, apart from GANs’ significant success, we detail the
failures of GANs due to their time complexity and unstable training. Although
GANs have been phenomenal for the generation of hyper-realistic data, current
progress in deep learning depicts an alternative narrative. Recently developed
architectures such as Diffusion models have demonstrated significant success
and outperformed GANs on image synthesis. On the other hand, Transformers, a
deep learning architecture based on a multi-head attention mechanism, has been
used within GAN architecture to enhance its performance. Furthermore, Large
Language Models, a widely utilized deep learning structure designed for
comprehending and producing natural language, have been incorporated into GAN
architecture to bolster its effectiveness. The hybridization of PINN and GAN
namely, PI-GAN can solve inverse and mixed stochastic problems based on a
limited number of scattered measurements. On the contrary, GANs’ ability which
relies on large data for training, using physical laws inside GANs in the form
of stochastic differential equations can mitigate the limited data problem.
Several hybrid approaches combining GAN with other powerful deep learners are
showing great merit and success as discussed in the discussion section.
Finally, several applications of GANs over the last decade are summarized and
criticized throughout the article.

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