1 00:00:00,690 --> 00:00:05,370 Now we are going to solve a 2D heat equation. 2 00:00:05,670 --> 00:00:11,910 The difference between 1D and 2D heat equation is simply in 2D. 3 00:00:11,940 --> 00:00:18,270 You have this additional parameter, which is the diffusion in the direction of Y. 4 00:00:18,780 --> 00:00:26,460 So the change of u with or the temperature with T will equal. 5 00:00:27,360 --> 00:00:38,460 If Alpha Square or let's say the whole thing is is just a factor that's going to be multiplied by the 6 00:00:38,460 --> 00:00:45,180 diffusion or the second derivative of U with respect to X and the second derivative of U with respect 7 00:00:45,180 --> 00:00:46,140 to Y. 8 00:00:46,560 --> 00:00:51,540 So again, this is the temporal form or the temporal. 9 00:00:52,500 --> 00:00:59,940 Well, you can say like component, which is a time based component or the change of U with respect 10 00:00:59,940 --> 00:01:02,640 to T and all of this. 11 00:01:02,640 --> 00:01:11,610 This one and this one is the spatial derivative or component, which is the derivative of U with respect 12 00:01:11,610 --> 00:01:17,250 to x and the second derivative of U with respect to x and the second derivative of U with respect to 13 00:01:17,250 --> 00:01:17,550 Y. 14 00:01:17,850 --> 00:01:25,440 Of course, U is the most the heat equation is usually the temperature, but it's it's basically describe 15 00:01:25,440 --> 00:01:30,340 a system that will have a diffusive property. 16 00:01:31,200 --> 00:01:32,270 So. 17 00:01:32,280 --> 00:01:33,900 So that's it. 18 00:01:33,900 --> 00:01:37,080 And this just the coefficient. 19 00:01:37,110 --> 00:01:39,750 Diffusion coefficient component. 20 00:01:39,900 --> 00:01:43,290 It can be different for every one of them. 21 00:01:43,290 --> 00:01:50,340 So we might have in the past I think it was K or A or V, I'm not sure. 22 00:01:50,340 --> 00:01:56,370 But basically this component can be different for the X and different for Y. 23 00:01:56,400 --> 00:02:05,020 Some materials will have a different coefficient regarding the diffusivity of or the heat transfer in 24 00:02:05,040 --> 00:02:07,470 the x direction is different than the y direction. 25 00:02:07,470 --> 00:02:08,370 So. 26 00:02:08,980 --> 00:02:12,280 These things will will be different. 27 00:02:12,490 --> 00:02:17,950 And but in this case, it will be taken very simple. 28 00:02:17,950 --> 00:02:20,440 Simply in this example. 29 00:02:20,440 --> 00:02:21,760 Actually, I'm going to I'm going to set it. 30 00:02:21,760 --> 00:02:28,180 One is just I don't really want to study its effect or work about its effect. 31 00:02:28,220 --> 00:02:37,690 The purpose of all of this is to simply explain how to solve heat equation, a 2D heat equation using 32 00:02:37,750 --> 00:02:41,110 um, this basically the um. 33 00:02:42,820 --> 00:02:45,250 A numeral neural neural networks. 34 00:02:45,250 --> 00:02:46,930 So we we do pens. 35 00:02:47,230 --> 00:02:49,180 So let's start. 36 00:02:49,180 --> 00:02:52,090 Let's start defining the needed libraries. 37 00:02:52,090 --> 00:02:56,650 And of course, um, the, the, the network. 38 00:02:56,680 --> 00:02:58,480 This is what we're going to do first. 39 00:02:58,480 --> 00:03:01,240 So the first thing is what we call it. 40 00:03:01,240 --> 00:03:03,460 Like we can say heat. 41 00:03:04,380 --> 00:03:05,640 Uh, to the. 42 00:03:06,630 --> 00:03:07,650 Into the. 43 00:03:10,180 --> 00:03:12,040 Bins mean main. 44 00:03:14,250 --> 00:03:18,480 This function and then we start importing the needed library. 45 00:03:18,510 --> 00:03:20,700 So import torch. 46 00:03:20,730 --> 00:03:21,840 Import. 47 00:03:23,250 --> 00:03:27,270 Import torch dot n. 48 00:03:27,270 --> 00:03:27,690 N. 49 00:03:27,690 --> 00:03:28,230 As n. 50 00:03:28,230 --> 00:03:28,770 N. 51 00:03:29,310 --> 00:03:29,970 Import. 52 00:03:29,970 --> 00:03:36,690 This time it will be very simple network and very simple optimization algorithms. 53 00:03:36,960 --> 00:03:42,240 We're going to just use Adam since just the out of the box very common. 54 00:03:42,900 --> 00:03:51,480 Adam And because the accuracy is not really what we are looking for, we're looking for mainly the concept 55 00:03:51,480 --> 00:03:54,930 and how can we do it like in a step by step manner. 56 00:03:54,930 --> 00:03:59,970 This is, this is what we want to this is why we took this course. 57 00:04:00,540 --> 00:04:09,900 So torch to Optim as Optim this one we will just we will take the Adam optimizer from it and see born 58 00:04:10,320 --> 00:04:22,410 as SNS for the we use it before for the heat plot plot lap matplotlib.pyplot. 59 00:04:22,980 --> 00:04:25,350 By plot as plt. 60 00:04:25,500 --> 00:04:27,570 The usual guy. 61 00:04:28,020 --> 00:04:38,220 The first thing we will do, we can call bins pens in n dot module, same as everything. 62 00:04:38,340 --> 00:04:45,870 We always inherit the a neural network module and init. 63 00:04:46,980 --> 00:04:47,640 Uh, sorry. 64 00:04:47,640 --> 00:04:50,760 This one and then self. 65 00:04:51,430 --> 00:04:53,620 And again, super. 66 00:04:54,930 --> 00:04:55,920 Pins. 67 00:04:57,050 --> 00:04:57,830 Bins. 68 00:04:58,950 --> 00:05:00,540 Self. 69 00:05:01,970 --> 00:05:02,600 Dot. 70 00:05:03,650 --> 00:05:04,610 In it. 71 00:05:06,430 --> 00:05:09,570 And now we will do the nut. 72 00:05:09,570 --> 00:05:18,340 Not always like the usual thing, which is actually building the network in dot sequential is capital 73 00:05:18,340 --> 00:05:22,660 letter seek when shell. 74 00:05:23,410 --> 00:05:27,450 And we start in n dot. 75 00:05:28,630 --> 00:05:29,290 Engineer. 76 00:05:30,900 --> 00:05:31,500 Lean. 77 00:05:32,640 --> 00:05:33,150 It. 78 00:05:34,360 --> 00:05:37,810 We will have a three and 64. 79 00:05:38,350 --> 00:05:41,050 This is we need to let a little bit relaxed. 80 00:05:42,780 --> 00:05:46,230 Linear 64 means now we have 2D. 81 00:05:46,260 --> 00:05:51,510 That means we have X, we have Y, and we have time. 82 00:05:51,810 --> 00:05:55,980 This is we didn't have it before, so this is why we have this one. 83 00:05:56,220 --> 00:06:00,320 And of course, of course we can do a. 84 00:06:01,380 --> 00:06:09,720 Complicated network, but will take it very simple and very just, very simple neural network, which 85 00:06:09,720 --> 00:06:14,550 is just three inputs expanding and then just. 86 00:06:16,200 --> 00:06:18,270 Collapsing onto one, which is you. 87 00:06:18,420 --> 00:06:27,330 So in a linear 364, I think like I like I like to write to, to, to write it like this. 88 00:06:28,270 --> 00:06:40,570 So here linear 64 and then an N dot the activation function ten inch and we close and then an n dot 89 00:06:41,410 --> 00:06:44,170 linear linear. 90 00:06:44,500 --> 00:06:51,580 And this time the activation function will be 64 expanding into another 64. 91 00:06:52,510 --> 00:06:54,980 I turn right here and this in. 92 00:06:56,370 --> 00:06:58,320 Dot ten. 93 00:06:58,650 --> 00:06:59,800 Etch Ten. 94 00:06:59,820 --> 00:07:00,660 Etch. 95 00:07:01,370 --> 00:07:07,670 And that I then itch and this time it will close. 96 00:07:08,440 --> 00:07:09,010 Linear. 97 00:07:11,580 --> 00:07:14,820 We'll get 64 and collapse onto one. 98 00:07:15,810 --> 00:07:19,710 Three will have to go to 64. 99 00:07:20,130 --> 00:07:26,570 The 64 has to go another 64 and then it will collapse into a. 100 00:07:27,680 --> 00:07:28,500 Two into one. 101 00:07:28,500 --> 00:07:31,350 So the neural network will be the first layer three. 102 00:07:31,400 --> 00:07:32,880 Second layer is 64. 103 00:07:33,000 --> 00:07:35,580 Third layer is going to be another 64. 104 00:07:35,580 --> 00:07:38,580 And then the last layer will be one. 105 00:07:39,370 --> 00:07:45,850 Which is what we're trying to predict, which is the the you forward. 106 00:07:47,410 --> 00:07:48,610 Self. 107 00:07:49,370 --> 00:07:50,390 And X. 108 00:07:51,520 --> 00:07:52,660 Return. 109 00:07:53,930 --> 00:07:55,760 Self dot net. 110 00:07:56,740 --> 00:07:58,060 And that's it. 111 00:07:58,490 --> 00:07:59,350 Then shift, enter. 112 00:07:59,890 --> 00:08:01,960 So that's it. 113 00:08:01,960 --> 00:08:03,490 This is how we. 114 00:08:05,070 --> 00:08:07,140 A define our network. 115 00:08:07,170 --> 00:08:10,200 Of course you can make it a little bit more. 116 00:08:11,520 --> 00:08:18,760 Complicated, but usually because all of these problems so far is very simple problems, I would say. 117 00:08:18,780 --> 00:08:27,480 So, of course, if you if you have a little bit more complex network, for example, you're solving 118 00:08:27,480 --> 00:08:28,620 Navier-Stokes equations. 119 00:08:28,620 --> 00:08:31,740 You might need to account more more than you might. 120 00:08:31,770 --> 00:08:34,140 You need to build a bigger network. 121 00:08:34,150 --> 00:08:35,730 It seems like it. 122 00:08:36,610 --> 00:08:37,110 So. 123 00:08:38,750 --> 00:08:41,840 However, we have only the heat equation and it is 2D. 124 00:08:41,840 --> 00:08:45,350 So we this is the main changes. 125 00:08:45,350 --> 00:08:48,430 We have three here input and this one as output. 126 00:08:48,440 --> 00:08:51,110 So let's next class we will. 127 00:08:52,620 --> 00:08:55,740 Define some of the losses. 128 00:08:55,740 --> 00:09:04,080 And just same thing, initial condition, boundary condition, everything like building the the whole 129 00:09:04,080 --> 00:09:04,680 system.