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After we decide both the initial conditions and boundary condition.

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What we will need to do is we will start defining the PDE.

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We we're trying to solve and basically the network and here with X deep, deep is these things are quite

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simple.

5
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So or like concise regarding the code.

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So def we write a function that will take x and.

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Will take six and.

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Why are you in our case and then d or just make it you.

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Due.

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With time we're going to be d d e dot grad dot jacobian.

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And then we take the value of you.

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We take the value of X and I equals.

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Zero.

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To J equals one.

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So this is going to be.

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Our first giacobbe.

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The second Giacobbe is due over.

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This is the second derivative and we do the same thing.

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Well, what?

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What will change is the method in which we are going to.

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Dig the great.

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The Great Forum is a Haitian.

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And we it will take you or it will give us.

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Well, the value of you giving, of course, X.

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And.

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We have I equals zero and.

27
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J equals or J equals zero.

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Because the function this one.

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Well, we can call it X, we can call it other thing like for example, um, comp space.

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Is comp space.

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So it's I think it's better to call it like this.

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So comp space.

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This is the you, the one we want to calculate and comp space is basically it will has it will have

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two components.

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The first component is a space which is a which is J equals zero means this space of zero and the J

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equals one is the time.

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So this is basically how it works.

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So comp is both X and Y, and then we have I equals one, which is only you already we have only we

39
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are calculating only one thing, which is you and has one component which is because it's scalar.

40
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Or like nuts, it's most likely it will be like a victor, but it has one value only.

41
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So we put it on the zero and we don't have one.

42
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So basically this is what the what what it means.

43
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And I think comp is better to write it this way.

44
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And the last thing is we need to put return.

45
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Well d d u over d y.

46
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We just put this way.

47
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So basically, um, we have du over dt we move all of this to here and we create something called residual.

48
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The residual has to equal zero.

49
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So.

50
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We created this residual by Du over DT minus.

51
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K multiplied.

52
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Do you?

53
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Over the x and we shift enter.

54
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The next thing we need to do is basically defining the data.

55
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So data.

56
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This is going to define the actual data that we are going to train.

57
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As we said, we have two types of losses.

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The the losses related to the the initial condition, boundary condition and the losses related with

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the whole domain that has to follow the physics.

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So we need to define in this case the data that we are going to use for training, both for the well,

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the initial condition, boundary condition, as well as the whole domain.

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So data equals dot data.

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A dot time PD.

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And we open parentheses.

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I don't I wanted to to put it here, but anyway, Jim.

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Time.

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And.

68
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We pass this one basically, and we need to pass boundary condition and initial condition, which we

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define here, boundary condition and way up.

70
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We define here the initial condition and.

71
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We also number.

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Of domain equals 2540.

73
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And.

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Number.

75
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These are the points that's going to be in the domain.

76
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Number of.

77
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Bone dry.

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Equals 80.

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I'm just checking the spelling.

80
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Boundary.

81
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Yeah.

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And.

83
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Number of initial.

84
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Points 160.

85
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Number of test equals to 2540.

86
00:06:08,120 --> 00:06:09,680
And that's it.

87
00:06:10,940 --> 00:06:12,560
So this is the data.

88
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That's it.

89
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It's very quite like, lovely.

90
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And sorry.

91
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2000.

92
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Yeah, it's.

93
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It's.

94
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It's just a warning.

95
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We're using this.

96
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We're using this.

97
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But point.

98
00:06:31,240 --> 00:06:32,830
Yeah, I think no problem.

99
00:06:32,830 --> 00:06:34,000
Yeah I think no problem.

100
00:06:34,450 --> 00:06:38,080
So net equals we now define the network.

101
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And also defining the network is quite simple.

102
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Dot in.

103
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In.

104
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Forward neural network.

105
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And we open it.

106
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We have input of two, which is X and T.

107
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Plus we have 20.

108
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A neuron with three layers with 20 neurons.

109
00:07:05,780 --> 00:07:10,100
And the last thing is going to be plus one.

110
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Again, this is the syntax of this of this library.

111
00:07:18,170 --> 00:07:19,550
We have to follow it.

112
00:07:20,360 --> 00:07:21,920
And then we have.

113
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Ten each.

114
00:07:25,750 --> 00:07:28,570
And this activation function, of course.

115
00:07:28,570 --> 00:07:30,730
And there is a glorot.

116
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Route.

117
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Normal.

118
00:07:37,420 --> 00:07:37,890
It's.

119
00:07:39,770 --> 00:07:40,880
Yeah, this one.

120
00:07:40,880 --> 00:07:49,070
So this one is related with initializing the network itself and or in this method it will initialize

121
00:07:49,070 --> 00:07:52,670
the network weights with a mean equals zero.

122
00:07:52,670 --> 00:07:57,860
So they will change randomly, but the mean of them it will be always zero.

123
00:07:58,570 --> 00:08:00,790
If we want to see the grid, our grid, let's.

124
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Let's see it.

125
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Let's plot dot figure.

126
00:08:06,020 --> 00:08:09,050
And let's see our grid, how it looks.

127
00:08:09,050 --> 00:08:11,990
And of course, there is no grid in pens.

128
00:08:12,020 --> 00:08:13,430
It's a mistress.

129
00:08:15,270 --> 00:08:15,780
The method.

130
00:08:16,710 --> 00:08:18,330
Size equals.

131
00:08:19,690 --> 00:08:21,460
Ten and eight.

132
00:08:23,570 --> 00:08:26,120
PLT dot scatter.

133
00:08:27,440 --> 00:08:29,170
Data dot.

134
00:08:30,440 --> 00:08:31,220
Train.

135
00:08:32,710 --> 00:08:34,330
Takes all.

136
00:08:35,320 --> 00:08:37,420
This is, of course, syntax.

137
00:08:37,450 --> 00:08:41,560
We want to have the points.

138
00:08:41,590 --> 00:08:43,600
We will print them.

139
00:08:43,780 --> 00:08:45,480
This is the X.

140
00:08:46,290 --> 00:08:51,120
And this is the equivalent values of Y.

141
00:08:52,490 --> 00:08:53,450
And of course.

142
00:08:55,580 --> 00:08:56,500
Dot x.

143
00:08:58,110 --> 00:08:58,860
Label.

144
00:08:59,930 --> 00:09:00,720
It's X.

145
00:09:04,820 --> 00:09:05,570
Why?

146
00:09:08,410 --> 00:09:10,240
And we destroy the thing.

147
00:09:10,330 --> 00:09:12,370
PLT dot show.

148
00:09:14,860 --> 00:09:16,480
So what does it mean?

149
00:09:17,050 --> 00:09:18,490
Of course it's random points.

150
00:09:18,490 --> 00:09:19,720
It's just separated.

151
00:09:19,720 --> 00:09:26,410
But the number of points in the in the whole boundaries is 80.

152
00:09:26,500 --> 00:09:29,170
This 80 and this is 80.

153
00:09:29,170 --> 00:09:35,280
And this we have a 160 points and all this in the domain.

154
00:09:35,290 --> 00:09:38,470
We have 2540 points.

155
00:09:38,680 --> 00:09:43,810
So basically, this is how our computation domain will look like.

156
00:09:43,810 --> 00:09:49,030
This is we we will have a kind of fine here, lots of points here.

157
00:09:49,030 --> 00:09:49,720
We will have.

158
00:09:49,720 --> 00:09:54,400
And I'm not so bad points in the boundary we have we will have a lot of boundary in the middle.

159
00:09:54,400 --> 00:09:57,760
And this is how we will calculate this is our actual domain.

160
00:09:58,350 --> 00:09:59,130
And.

161
00:09:59,670 --> 00:10:00,210
Yeah.

162
00:10:00,210 --> 00:10:00,860
So that's it.

163
00:10:00,870 --> 00:10:01,410
That's it.

164
00:10:01,410 --> 00:10:09,780
And let's next time we will solve this like the heat equation.

165
00:10:09,780 --> 00:10:12,210
And of course later we will see the results.