<s>
Physics-informed	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
(	O
PINNs	O
)	O
are	O
a	O
type	O
of	O
universal	O
function	O
approximators	O
that	O
can	O
embed	O
the	O
knowledge	O
of	O
any	O
physical	O
laws	O
that	O
govern	O
a	O
given	O
data-set	O
in	O
the	O
learning	O
process	O
,	O
and	O
can	O
be	O
described	O
by	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
.	O
</s>
<s>
The	O
prior	O
knowledge	O
of	O
general	O
physical	O
laws	O
acts	O
in	O
the	O
training	O
of	O
neural	B-Architecture
networks	I-Architecture
(	O
NNs	O
)	O
as	O
a	O
regularization	O
agent	O
that	O
limits	O
the	O
space	O
of	O
admissible	O
solutions	O
,	O
increasing	O
the	O
correctness	O
of	O
the	O
function	O
approximation	O
.	O
</s>
<s>
This	O
way	O
,	O
embedding	O
this	O
prior	O
information	O
into	O
a	O
neural	B-Architecture
network	I-Architecture
results	O
in	O
enhancing	O
the	O
information	O
content	O
of	O
the	O
available	O
data	O
,	O
facilitating	O
the	O
learning	O
algorithm	O
to	O
capture	O
the	O
right	O
solution	O
and	O
to	O
generalize	O
well	O
even	O
with	O
a	O
low	O
amount	O
of	O
training	O
examples	O
.	O
</s>
<s>
However	O
,	O
these	O
equations	O
cannot	O
be	O
solved	O
exactly	O
and	O
therefore	O
numerical	B-Algorithm
methods	O
must	O
be	O
used	O
(	O
such	O
as	O
finite	B-Algorithm
differences	I-Algorithm
,	O
finite	B-Application
elements	I-Application
and	O
finite	B-Algorithm
volumes	I-Algorithm
)	O
.	O
</s>
<s>
Recently	O
,	O
solving	O
the	O
governing	O
partial	O
differential	O
equations	O
of	O
physical	O
phenomena	O
using	O
deep	B-Algorithm
learning	I-Algorithm
has	O
emerged	O
as	O
a	O
new	O
field	O
of	O
scientific	O
machine	O
learning	O
(	O
SciML	O
)	O
,	O
leveraging	O
the	O
universal	O
approximation	O
and	O
high	O
expressivity	O
of	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
In	O
general	O
,	O
deep	O
neural	B-Architecture
networks	I-Architecture
could	O
approximate	O
any	O
high-dimensional	O
function	O
given	O
that	O
sufficient	O
training	O
data	O
are	O
supplied	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
physics-informed	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
(	O
PINNs	O
)	O
leverage	O
governing	O
physical	O
equations	O
in	O
neural	B-Architecture
network	I-Architecture
training	O
.	O
</s>
<s>
In	O
this	O
fashion	O
,	O
a	O
neural	B-Architecture
network	I-Architecture
can	O
be	O
guided	O
with	O
training	O
data	O
that	O
do	O
not	O
necessarily	O
need	O
to	O
be	O
large	O
and	O
complete	O
.	O
</s>
<s>
PINNs	O
allow	O
for	O
addressing	O
a	O
wide	O
range	O
of	O
problems	O
in	O
computational	O
science	O
and	O
represent	O
a	O
pioneering	O
technology	O
leading	O
to	O
the	O
development	O
of	O
new	O
classes	O
of	O
numerical	B-Algorithm
solvers	O
for	O
PDEs	O
.	O
</s>
<s>
In	O
addition	O
,	O
they	O
allow	O
for	O
exploiting	O
automatic	B-Algorithm
differentiation	I-Algorithm
(	O
AD	O
)	O
to	O
compute	O
the	O
required	O
derivatives	O
in	O
the	O
partial	O
differential	O
equations	O
,	O
a	O
new	O
class	O
of	O
differentiation	O
techniques	O
widely	O
used	O
to	O
derive	O
neural	B-Architecture
networks	I-Architecture
assessed	O
to	O
be	O
superior	O
to	O
numerical	B-Algorithm
or	O
symbolic	B-Algorithm
differentiation	I-Algorithm
.	O
</s>
<s>
and	O
approximating	O
by	O
a	O
deep	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
This	O
network	O
can	O
be	O
differentiated	O
using	O
automatic	B-Algorithm
differentiation	I-Algorithm
.	O
</s>
<s>
and	O
approximating	O
by	O
a	O
deep	O
neural	B-Architecture
network	I-Architecture
,	O
results	O
in	O
a	O
PINN	O
.	O
</s>
<s>
This	O
network	O
can	O
be	O
derived	O
using	O
automatic	B-Algorithm
differentiation	I-Algorithm
.	O
</s>
<s>
Piece-wise	O
approximation	O
has	O
been	O
an	O
old	O
practice	O
in	O
the	O
field	O
of	O
numerical	B-Algorithm
approximation	O
.	O
</s>
<s>
XPINNs	O
is	O
a	O
generalized	O
space-time	O
domain	O
decomposition	O
approach	O
for	O
the	O
physics-informed	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
(	O
PINNs	O
)	O
to	O
solve	O
nonlinear	O
partial	O
differential	O
equations	O
on	O
arbitrary	O
complex-geometry	O
domains	O
.	O
</s>
<s>
Compared	O
to	O
PINN	O
,	O
the	O
XPINN	O
method	O
has	O
large	O
representation	O
and	O
parallelization	O
capacity	O
due	O
to	O
the	O
inherent	O
property	O
of	O
deployment	O
of	O
multiple	O
neural	B-Architecture
networks	I-Architecture
in	O
the	O
smaller	O
subdomains	O
.	O
</s>
<s>
This	O
drawback	O
is	O
overcome	O
by	O
using	O
functional	O
interpolation	O
techniques	O
such	O
as	O
the	O
Theory	O
of	O
Functional	O
Connections	O
(	O
TFC	O
)	O
'	O
s	O
constrained	O
expression	O
,	O
in	O
the	O
Deep-TFC	O
framework	O
,	O
which	O
reduces	O
the	O
solution	O
search	O
space	O
of	O
constrained	O
problems	O
to	O
the	O
subspace	O
of	O
neural	B-Architecture
network	I-Architecture
that	O
analytically	O
satisfies	O
the	O
constraints	O
.	O
</s>
<s>
A	O
further	O
improvement	O
of	O
PINN	O
and	O
functional	O
interpolation	O
approach	O
is	O
given	O
by	O
the	O
Extreme	O
Theory	O
of	O
Functional	O
Connections	O
(	O
X-TFC	O
)	O
framework	O
,	O
where	O
a	O
single-layer	O
Neural	B-Architecture
Network	I-Architecture
and	O
the	O
extreme	B-Algorithm
learning	I-Algorithm
machine	I-Algorithm
training	O
algorithm	O
are	O
employed	O
.	O
</s>
<s>
In	O
fact	O
,	O
instead	O
of	O
using	O
a	O
simple	O
fully	O
connected	O
neural	B-Architecture
network	I-Architecture
,	O
PIPN	O
uses	O
PointNet	O
as	O
the	O
core	O
of	O
its	O
neural	B-Architecture
network	I-Architecture
.	O
</s>
<s>
PointNet	O
has	O
been	O
primarily	O
designed	O
for	O
deep	B-Algorithm
learning	I-Algorithm
of	O
3D	O
object	O
classification	O
and	O
segmentation	O
by	O
the	O
research	O
group	O
of	O
Leonidas	O
J	O
.	O
Guibas	O
.	O
</s>
