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What we need to do now is to define the PD of Navier Stokes equation.

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Of course, our PD is the Navier Stokes equation, which is.

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This one.

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So we will do it.

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And of course, a step by step.

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Okay.

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So to define it, we need as we did before, it will have.

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X and Y.

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And X is basically the values in which we are.

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The independent values and the Y is the dependent values.

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So X is is like like x and y and y is u, P and V, of course.

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So the first thing is we have in this equation, let's see, we have we have we have second derivative

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of u over all of them has to be defined.

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So let's start with d, u, d, x, d, x equals.

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And we will see.

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Of course there are.

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The good thing is in the documentation of deep, this equation is more or less similar to Navier-Stokes

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equation.

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And then we can see how they are basically doing this like first derivative, this is the second derivative

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and so on.

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So the first derivative and I want you to also kind of training is you always look at the documentation,

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you always try to mimic the documentation and even you use another method or another library to to do

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this work.

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You can always use it.

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And for example, for the first derivative, you use this one and you just copy it and put it here.

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D Grad dot Jacobian.

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What is the first?

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Of course u is not you here put it he's he's putting u here and x we have here u is y so we put y changing

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with x x he put it here, small x we put a with big x and zero zero means the first component is the

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U, which is basically the U that he put here.

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Y So that means is means the velocity which is zero and J is X, which is also zero.

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So d v over d, x has to be a little bit different.

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The V means the velocity.

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Or d u over d y d u over d y.

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We need velocity, which is zero, but J has to be one.

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Why?

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Because now we are talking about.

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A y axis.

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So here you also see it.

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He's doing the same thing.

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Velocity and Y has to be J equals one.

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Okay.

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Okay, So we continue.

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De.

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V over and over.

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D y, d v over will become one zero and over.

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D y is one.

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Y y.

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Because V is one.

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The second component.

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The first component is zero.

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U second component is one V and of course, J.

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As we go.

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The last thing is or not the last thing.

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The first derivative.

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Last thing is we need.

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P.

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And P over the X and over Y.

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And here is going to be two and two zero for X and one for Y.

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Of course, you can also see it in here.

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The P over the X you.

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Just like 2021.

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So it's all, uh.

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Yeah, this one is not exist.

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So basically, this is our first derivative, and this is just we see a little bit the equation.

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We can see we have the U over the X, we have the U over D y, we have DV over the x over d, y, d,

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p, over the x, p over d y.

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So now we already made this.

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Made this, this, this, this and this.

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Now, the second derivative is a bit tricky.

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Tricky because it's, it's, it seems a strange logic the way they define it.

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But basically the second derivative, which is this one, um, or let's take this 1UXX the first thing

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he defined the component which is the U and then it will become, which is the first derivative and

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what is the second derivative.

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So.

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D u over x x well equals we use heisen.

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Or Haitian.

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A method.

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And here it's going to be why?

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Here.

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Why?

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Here.

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X component is you.

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It is zero.

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X is also zero and Y is also zero.

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So this is the way to go.

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Now we.

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Do another thing, which is the second derivative of U over Y.

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The first component is zero.

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That's correct.

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The second component is one, and J is one.

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Why?

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Because first derivative.

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The second derivative of Y and y.

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So this is y y it will be one one.

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Do the same thing, but this time dv.

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Second derivative dv.

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DV.

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We have the same thing here.

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X x will be x x and.

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But here one.

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And when.

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That's it.

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That's the small derivatives.

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However, we need now to combine everything.

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D or E?

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Of.

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The you.

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This is the momentum.

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Or.

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Yeah, well you can say like this of the.

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Which is that the momentum in the direction and.

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Continuity.

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Continuity.

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We need to all be defined.

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So the first one is we need to consider, well, in our case you do over the X, so let's consider this

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equation.

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D u over d x multiplied by u.

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So which is why this value is going to be everything and 0 to 1 which is the first component multiplied

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d u d x plus.

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The same thing.

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Well for all the points.

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And zero one bit became one too, because now it's you.

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Multiplied the.

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You over the one thing.

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The over the.

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The you over.

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Why?

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Plus one over Rho.

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Or roll.

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Multiply DPI.

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Over.

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Minus.

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Mu over rho.

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Multiplied.

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The you.

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X.

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X plus do.

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Why?

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Why?

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What does that mean?

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Means.

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This is you.

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The you over the and this is the the you over.

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And this one.

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We have to move it to this direction.

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One over P the P over this plus become minus.

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And this is we said the kinematic viscosity, which is mu over rho.

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So this is we put it mu over rho the second derivative of u over the second derivative of u over D This

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is why this become minus and this become positive.

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This is going to be the first residual.

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The second thing, what we will do here, where more or less the same you and here V but this is becomes

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the V over the y, the v over y, dp over y.

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And.

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Second derivative of.

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Y y.

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Oh, no, it's just here.

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V not because X and Y already exist in both.

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So here you and you.

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I think it's pretty much no problem.

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The last one is.

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Well, there's this very simple thing.

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Is this one.

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The U over x plus D over over d, y, d.

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You're over X.

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Plus DV.

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Why?

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Return.

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Everything.

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Just these.

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These are the things that has to be equal.

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That has to equal to zero because.

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We have all of this.

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It's called residual and residual.

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Basically, we keep all the equation on one side and we equal it to zero.

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It has to be achieved if we want to converge our solution.

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Shift.

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Enter.

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Hopefully we didn't forget anything.

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And this is basically our now data.

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The data.

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We need to create.

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Data is basically that the neural network has to train on it in order to converge.

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The dot data dot PD.

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What we need to add.

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The first one is geometry.

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This one.

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Second one is.

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Which is the whole thing, this one.

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And of course, it has these little beads.

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I would say like the like everything it will have now, number of domain points equals one 2000.

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Domain points, number of boundary.

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Bone.

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The points is again 2000.

195
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This should be domain boundary is also okay.

196
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Number of test.

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Just put it wherever.

198
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Just keep it.

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100.

200
00:12:40,610 --> 00:12:41,540
Just 200.

201
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Because.

202
00:12:42,740 --> 00:12:47,840
Follow the 2 to 2 and pretty much.

203
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That's it.

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Shift.

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Enter.

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And.

207
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There's the problem.

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It is business.

209
00:13:01,650 --> 00:13:02,970
Oh, sorry, sorry, sorry, sorry.

210
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Boundary conditions.

211
00:13:04,470 --> 00:13:05,670
We forgot.

212
00:13:07,070 --> 00:13:08,300
It's a big problem.

213
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Sorry.

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00:13:11,680 --> 00:13:12,720
Here.

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00:13:15,380 --> 00:13:17,120
We put our.

216
00:13:26,220 --> 00:13:27,300
Well, it's not.

217
00:13:27,300 --> 00:13:27,700
Yeah.

218
00:13:28,410 --> 00:13:32,310
The first one is, well, the wall boundary.

219
00:13:33,710 --> 00:13:36,440
You wall boundary.

220
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The.

221
00:13:39,490 --> 00:13:40,540
And.

222
00:13:41,420 --> 00:13:42,440
And let's.

223
00:13:45,190 --> 00:13:47,170
And let the.

224
00:13:49,710 --> 00:13:50,430
Pressure.

225
00:13:55,250 --> 00:13:59,150
Um, pressure outlet v and outlet pressure.

226
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I think now it seems correct.

227
00:14:03,140 --> 00:14:03,740
This is.

228
00:14:03,740 --> 00:14:05,780
Let's just skipping a bit.

229
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Higher.

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00:14:08,790 --> 00:14:10,190
Seems okay.

231
00:14:10,620 --> 00:14:10,850
Okay.

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00:14:10,860 --> 00:14:12,510
Now I think it will work.

233
00:14:13,540 --> 00:14:13,890
Shift.

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00:14:13,900 --> 00:14:14,350
Enter.

235
00:14:14,380 --> 00:14:25,030
Okay, so now he's passing all this data to our functions and he's checking if it's if these points.

236
00:14:25,450 --> 00:14:26,950
This is a problem.

237
00:14:26,950 --> 00:14:27,700
There are.

238
00:14:33,280 --> 00:14:36,330
It just seems to be some issue.

239
00:14:43,580 --> 00:14:47,420
The problem is we didn't pass on X and on boundary.

240
00:14:47,420 --> 00:14:49,070
We didn't pass the points.

241
00:14:49,070 --> 00:14:51,020
So this is what causes trouble.

242
00:14:51,050 --> 00:14:52,160
Now we.

243
00:14:52,520 --> 00:14:59,300
Well, it's better to shift like run the whole thing from the beginning because we don't know what we

244
00:14:59,300 --> 00:15:00,500
already defined.

245
00:15:01,640 --> 00:15:03,530
I think now is okay.

246
00:15:03,530 --> 00:15:09,620
And before I put here 2000 points, now it's just change it to 200 points.

247
00:15:10,940 --> 00:15:11,710
It seems.

248
00:15:11,710 --> 00:15:12,310
No.

249
00:15:13,740 --> 00:15:15,300
Point is running.

250
00:15:15,300 --> 00:15:16,470
Okay.

251
00:15:18,080 --> 00:15:20,450
Let's take just a little bit like this.

252
00:15:22,690 --> 00:15:23,230
Yeah.

253
00:15:24,760 --> 00:15:25,240
Now.

254
00:15:25,240 --> 00:15:25,690
It's.

255
00:15:25,690 --> 00:15:26,560
It's good.

256
00:15:26,620 --> 00:15:27,760
Do you see this?

257
00:15:27,910 --> 00:15:32,470
False and true means we create a 200 boundaries.

258
00:15:32,770 --> 00:15:34,330
We create 200 boundary points.

259
00:15:34,330 --> 00:15:37,570
If the boundary matches the well.

260
00:15:37,600 --> 00:15:44,680
True or false, the the the boundary condition here we put it will give false and it will not apply

261
00:15:44,920 --> 00:15:49,450
this boundary condition with zero or UN or whatever we define.

262
00:15:49,450 --> 00:15:53,410
So this is a little bit more solid way to define it rather than before.

263
00:15:53,410 --> 00:15:59,920
However, before method can work, however, you have to be very cautious that when you enter this boundary

264
00:15:59,920 --> 00:16:02,740
points you have to decide which is which.

265
00:16:04,450 --> 00:16:07,140
So now we have done it.

266
00:16:07,150 --> 00:16:09,610
Let's just plot.

267
00:16:11,470 --> 00:16:12,190
The.

268
00:16:13,110 --> 00:16:13,710
Figure.

269
00:16:16,670 --> 00:16:18,380
Like plot the point itself.

270
00:16:18,980 --> 00:16:20,240
Fixed size.

271
00:16:22,150 --> 00:16:23,320
Equals.

272
00:16:24,750 --> 00:16:25,470
Eight.

273
00:16:29,210 --> 00:16:29,660
Dot.

274
00:16:31,100 --> 00:16:32,350
Scatter.

275
00:16:40,330 --> 00:16:44,220
And that train is the same as we did before.

276
00:16:45,260 --> 00:16:53,420
Uh, so just this is the X, and we do the same thing here.

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Or why?

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X equals 0.5.

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Well.

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Pldt.

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That X label.

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And just plotting things.

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Yep, that's it.

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PLT dot show.

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So these are our points.

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We can see this is the boundary points.

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This is the domain points.

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Of course, this is like.

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Scattered randomly and.

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The last thing we have to do is define the neural network.

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So.

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The network will equals the dot maps dot forward neural network and it will have input of two Y is two

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because we have X and y only.

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A plus.

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We have 64 multiplied by multiplied five.

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This is going to be our hidden layer.

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We put it like this.

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And output is three, Y is three because u, V and P3A activation function.

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Then it.

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Then H and the initialization we said we need to initialize everything with a mean of zero.

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So they will spread, but they will have the mean of zero.

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Elk Lodge.

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Rot uniform.

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Form.

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Shift into.

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And this is our network.

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Next thing, what we will do is we need to train the actual network.