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Pins or physics informed neural networks are.

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A neural network that is a deep neural network that combines both the power of neural networks with

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the physics laws.

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Of course, the purpose of doing pins is to solve a mainly partial differential equations.

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However, the idea is is mainly about.

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Using physics and its laws along with neural networks, to achieve modeling, modeling of a system.

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And by that we can actually calculate a whole system in a very quick way.

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It's all about how we train the neural network and for how many boundary condition, for how many initial

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conditions to actually make it successful to achieve that.

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So basically this is a pens and as I said, it is a combination physics with neural networks.

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Now how it works.

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So basically the pins will work using or training on a data and this data or in order to model an actual

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system, we need to do well, mainly two things.

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The first thing is we need to to have this network and have this loss to be calculated.

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Now, it's not enough to actually calculating the the the loss.

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That is, we model the system as X, for example, if it's one dimensional system, we have X and T

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and we will get you.

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It's not enough to have an input of X, an input of T, and we only get you for let's say, the boundary

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condition and so on.

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And of course the initial condition.

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The problem is what we will face is there are many points in the domain like in between that needs to

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converge.

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And this is why we need well, basically the the partial differential equation.

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And these points have to follow these pdes laws.

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And well, basically, now we have the we have to we have the loss that has to be minimized.

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And the loss comes from two sources.

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That is the PDE and of course the partial well, the the the boundary condition and the initial condition.

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So basically this is what how how mainly it works.

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And so we need a system.

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We need a partial differential equation that is capable to solve our problem.

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Capable is not it depends on how complex the problem.

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We will build these networks.

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And it's for for simple equations.

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We don't have to be very deep.

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And of course we we don't have to have many input conditions and of course, output conditions.

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The second thing, of course, we need a that describes the physical systems and then we just calculate

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the loss.

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So, so this is the idea behind pens is just need to have the PDE to calculate the losses.

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Now how the domain works or okay, you want to calculate the loss, but for what?

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Where did you where do you get the data?

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The data is actually is the boundary condition and the initial condition.

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So this is you have two parts of data.

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The first one is boundary condition and initial condition.

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This is these conditions are part of the data and it's very important part of the data.

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The second part of the data is the domain parts.

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And these domain points which are in the middle all over here randomly, you know, you can put them

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over there just randomly with the with the sufficient density.

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Of course, it has to be sufficient.

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And the sufficient means.

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It depends on the actual equation and the problem we're trying to solve.

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And we have to basically calculate the loss in order to satisfy the boundary conditions and satisfy

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the physics of the initial conditions.

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So the the network, if you put a a point in the boundary, the network has to understand this is this

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is a boundary point and it has a specific value in order to achieve so well, we need to.

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That means we have to create many boundary points and to feed these boundary points to the network in

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order to make sure that whenever the network sees this as a boundary point, it will give it the correct

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answer.

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The domain point is the same.

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We put a domain point, and this domain point has, along with everything around it, has to satisfy

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the physics law.

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We keep doing this until basically the network is converged and the loss is minimized.

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Getting, well, a converged solution, get a solution.

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And then once we have that, we have the network, we give it X, we give it T or x, Y, whatever the

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network, whatever the situation is, and it will give us the answer.

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So how to solve it?

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We have basically a general steps.

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The first step is you define the network or this is how to solve it while using torch or using, you

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know, like a library that is will handle the deep neural network.

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However, of course we will also solve it using some easy or well written libraries for only solving

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pens.

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However, these are the steps that we need to achieve in order to solve the network using, um, well,

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mainly torch.

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The first thing is we need to define the network.

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We need to define what are the inputs and what are the outputs.

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For example, if it's a 2D problem, we will have X and Y and we will get a value or multiple value.

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For example, the velocity in the direction, the velocity in the V direction and let's say the rho,

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the the density.

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After that, we have to define the initial condition and the boundary conditions and a well followed

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by this.

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We have to set our optimizers, usually in pins.

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We start with a kind of general optimizer like Adam, and then we go for a optimizers that will take

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into consideration or will tweak the loss to be as small as possible.

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And then what we will do is we need to define the loss function.

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We're trying to define the loss functions to basically satisfy the both the boundary, the the the the

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the boundary conditions and the initial condition as well as the residual errors or the pdes that needs

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to be calculated.

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And after that we just run the training loop we need, same as any network.

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It has epochs, it has to be trained and, you know, like go round and round until the loss is minimized.

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And then once we have the result, we have the post-processing, we get these results and we plot them

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as any CFD or finite difference methods.

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So these are the general steps of solving pens.

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And hopefully in this course we will as much as possible be detailed in the explanation.

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Of course, pens and numerical solutions is not, well let's say, established.

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It's not well developed.

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But I think in the next in the in the future, we will have more and more well researchers and engineers

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that they are going to use it in order to get a fast solutions.