In[]:=
meshNetToExpression[depth_Integer,activationFunction_]:=With[{n=2depth,act=activationFunction},​​Block[{w=1,b=1},​​Nest[​​act[Total[#*Table[Subscript[W,w++],2]]+Subscript[B,b++]]&/@Partition[ArrayPad[#,{1,1}],2,1]&​​,{x}​​,n​​][[depth+1]]​​]​​]
In[]:=
meshNetToExpression[1,ϕ]
Out[]=
ϕ[
B
4
+
W
7
ϕ[
B
1
+x
W
2
]+
W
8
ϕ[
B
2
+x
W
3
]]
In[]:=
meshNetToExpression[2,ϕ]
Out[]=
ϕ[
B
12
+
W
23
ϕ[
B
7
+
W
13
ϕ[
B
3
+
W
6
ϕ[
B
1
+x
W
2
]]+
W
14
ϕ[
B
4
+
W
7
ϕ[
B
1
+x
W
2
]+
W
8
ϕ[
B
2
+x
W
3
]]]+
W
24
ϕ[
B
8
+
W
15
ϕ[
B
4
+
W
7
ϕ[
B
1
+x
W
2
]+
W
8
ϕ[
B
2
+x
W
3
]]+
W
16
ϕ[
B
5
+
W
9
ϕ[
B
2
+x
W
3
]]]]
In[]:=
meshNetToExpression[2,Ramp]
Out[]=
Ramp[
B
12
+Ramp[
B
7
+Ramp[
B
3
+Ramp[
B
1
+x
W
2
]
W
6
]
W
13
+Ramp[
B
4
+Ramp[
B
1
+x
W
2
]
W
7
+Ramp[
B
2
+x
W
3
]
W
8
]
W
14
]
W
23
+Ramp[
B
8
+Ramp[
B
4
+Ramp[
B
1
+x
W
2
]
W
7
+Ramp[
B
2
+x
W
3
]
W
8
]
W
15
+Ramp[
B
5
+Ramp[
B
2
+x
W
3
]
W
9
]
W
16
]
W
24
]
In[]:=
FullSimplify[%]
In[]:=
Simplify[meshNetToExpression[1,(Exp[#]-1)/(Exp[#]+1)&]]
Out[]=
-1+
B
4
+
-1+
B
1
+x
W
2


W
7
1+
B
1
+x
W
2

+
-1+
B
2
+x
W
3


W
8
1+
B
2
+x
W
3


1+
B
4
+
-1+
B
1
+x
W
2


W
7
1+
B
1
+x
W
2

+
-1+
B
2
+x
W
3


W
8
1+
B
2
+x
W
3


In[]:=
Series[%,{x,0,1}]
Out[]=
-1+
B
4
+
-1+
B
1


W
7
1+
B
1

+
-1+
B
2


W
8
1+
B
2


1+
B
4
+
-1+
B
1


W
7
1+
B
1

+
-1+
B
2


W
8
1+
B
2


+
4
B
4
+
-1+
B
1


W
7
1+
B
1

+
-1+
B
2


W
8
1+
B
2



B
1

W
2
W
7
+2
B
1
+
B
2

W
2
W
7
+
B
1
+2
B
2

W
2
W
7
+
B
2

W
3
W
8
+2
B
1
+
B
2

W
3
W
8
+
2
B
1
+
B
2

W
3
W
8
x
2
(1+
B
1

)
2
(1+
B
2

)
2
1+
B
4
+
-1+
B
1


W
7
1+
B
1

+
-1+
B
2


W
8
1+
B
2


+
2
O[x]
In[]:=
Rotate[GridGraph[{5,5}],45Degree]
Out[]=
In[]:=
meshNetToExpression[2,ϕ]/.Subscript[B,_]->0
Out[]=
ϕ[
W
23
ϕ[
W
13
ϕ[
W
6
ϕ[x
W
2
]]+
W
14
ϕ[
W
7
ϕ[x
W
2
]+
W
8
ϕ[x
W
3
]]]+
W
24
ϕ[
W
16
ϕ[
W
9
ϕ[x
W
3
]]+
W
15
ϕ[
W
7
ϕ[x
W
2
]+
W
8
ϕ[x
W
3
]]]]
In[]:=
ExpressionTree[%]
Out[]=