1 00:00:00,690 --> 00:00:07,800 So now we will start doing the the kind of the training loop we start writing it. 2 00:00:07,800 --> 00:00:11,040 But of course this time also the optimizer. 3 00:00:11,040 --> 00:00:15,840 I as I said it will I will use quite simple optimizer which is Adam. 4 00:00:16,470 --> 00:00:17,430 So. 5 00:00:18,510 --> 00:00:19,560 Let's write it. 6 00:00:19,560 --> 00:00:28,560 Let's write the basically the training loop, but will leave a little bit here space because I want 7 00:00:28,560 --> 00:00:37,170 to put the loss function in here because in this code, other than in the past code, the things used 8 00:00:37,170 --> 00:00:39,300 to be a little bit messy. 9 00:00:39,300 --> 00:00:46,140 I would say here it's a little bit more organized into functions that in your case, if you want to 10 00:00:46,140 --> 00:00:52,680 change the the future, how these things will look, you can simply do that. 11 00:00:52,680 --> 00:00:54,060 So just. 12 00:00:55,040 --> 00:00:58,400 So this will be the the actual description of PDA. 13 00:00:58,700 --> 00:01:01,550 So here, let's do the. 14 00:01:02,470 --> 00:01:07,340 Optimization part to follow kind of the same methodology. 15 00:01:07,520 --> 00:01:13,730 And of course we will write the the main training loop, but this time is a little bit more organized, 16 00:01:13,730 --> 00:01:17,510 I would say, rather than the 1D situation. 17 00:01:17,810 --> 00:01:23,240 Now here a train with pin underscore. 18 00:01:23,630 --> 00:01:25,100 Underscore pin. 19 00:01:25,780 --> 00:01:28,330 And model. 20 00:01:29,360 --> 00:01:34,760 It will take and number of iterations. 21 00:01:35,580 --> 00:01:40,890 Integrations and number of points. 22 00:01:42,780 --> 00:01:47,220 Optimizer to miser equals optimum. 23 00:01:48,220 --> 00:01:48,940 Dot. 24 00:01:49,150 --> 00:01:49,900 Adam. 25 00:01:50,880 --> 00:01:56,190 And model dot parameters. 26 00:01:57,750 --> 00:02:03,930 With a learning rate of one e to the power minus three. 27 00:02:06,470 --> 00:02:08,690 I will just check it a little bit. 28 00:02:08,720 --> 00:02:10,320 Parameters. 29 00:02:10,340 --> 00:02:12,700 Parameters Here. 30 00:02:13,310 --> 00:02:13,750 There's. 31 00:02:13,790 --> 00:02:14,000 Yeah. 32 00:02:14,630 --> 00:02:15,170 Okay. 33 00:02:17,150 --> 00:02:18,560 That's it for the optimizer. 34 00:02:18,560 --> 00:02:27,740 However, we just need to define the the the points as we did before, just a little bit more for this 35 00:02:27,740 --> 00:02:28,190 step. 36 00:02:28,190 --> 00:02:31,640 And later, we will focus only on calculating the losses. 37 00:02:31,880 --> 00:02:38,600 So iteration in range of well number of iterations. 38 00:02:40,160 --> 00:02:42,220 Number of iterations. 39 00:02:44,610 --> 00:02:45,810 Optimizer. 40 00:02:47,430 --> 00:02:49,440 Is zero grid. 41 00:02:50,350 --> 00:02:52,270 Zero grad. 42 00:02:52,300 --> 00:02:53,350 Grad not great. 43 00:02:54,810 --> 00:02:55,410 Grant. 44 00:02:57,160 --> 00:03:00,430 And now we will generate the training data. 45 00:03:00,430 --> 00:03:02,260 And these are the training data. 46 00:03:02,260 --> 00:03:05,380 That means they are the points in the domain. 47 00:03:05,530 --> 00:03:09,040 These are points in the domain, not on the boundary. 48 00:03:09,160 --> 00:03:12,370 They might get in the boundary, but we don't really care because it's. 49 00:03:12,550 --> 00:03:20,050 Well yeah, I need to check but basically it they might even if they are some points that is generated 50 00:03:20,050 --> 00:03:27,460 randomly on the boundary, it will not cause a problem because we are generating a good amount of boundary 51 00:03:27,550 --> 00:03:28,300 points. 52 00:03:28,300 --> 00:03:34,920 But I think in this case, like when we have the points, it will I think it will generate from 0 to 53 00:03:34,920 --> 00:03:36,190 1 these points. 54 00:03:36,190 --> 00:03:41,950 So there is there are some a bit of risk, but but we don't really care because. 55 00:03:42,570 --> 00:03:43,860 We have boundary condition. 56 00:03:43,860 --> 00:03:49,860 And even if there are one training case that is a little bit strange, it will kind of. 57 00:03:50,820 --> 00:03:55,650 A be modified and even even this boundary condition. 58 00:03:57,010 --> 00:04:00,530 Well, it's about around that it will be zero. 59 00:04:00,540 --> 00:04:03,090 So and the boundary condition is already zero. 60 00:04:03,090 --> 00:04:09,090 So even this function, this boundary, it's think its value is near to zero. 61 00:04:09,810 --> 00:04:19,740 But yeah, it's if you if you have, um, well, just be careful about the points inside the boundary 62 00:04:19,770 --> 00:04:25,500 is different than the points in the boundary that these points are. 63 00:04:26,720 --> 00:04:32,350 At the point in the domain which is here, which we are generating random points in X, Y and T. 64 00:04:33,330 --> 00:04:41,550 And here we will just put it like generate this and we pass the number of points. 65 00:04:42,420 --> 00:04:43,320 Number of points. 66 00:04:43,770 --> 00:04:46,080 So that's it. 67 00:04:46,580 --> 00:04:53,390 That's the the that's the point that is in the domain that the need that we need to calculate. 68 00:04:53,390 --> 00:05:01,850 And then later I don't want to like too much, but basically we will use them to calculate the the loss 69 00:05:01,850 --> 00:05:07,850 which is we said there is a data loss which basically from the boundary condition and the initial condition 70 00:05:08,030 --> 00:05:17,990 and they have another loss that is generated from the actual, um, the points to follow the physics 71 00:05:17,990 --> 00:05:22,430 laws, which is the we have the 2D heat equation. 72 00:05:23,450 --> 00:05:23,900 Okay. 73 00:05:23,900 --> 00:05:28,940 So these are the points in the domain and now we have the points in the boundary. 74 00:05:28,940 --> 00:05:38,420 So y underscore b x underscore b t underscore b equals generate training data. 75 00:05:38,540 --> 00:05:40,190 Actually just take it all of them. 76 00:05:40,340 --> 00:05:44,360 This one and of course, points we have to put it here. 77 00:05:46,680 --> 00:05:47,550 And that's it. 78 00:05:48,910 --> 00:05:50,980 So these are now we created. 79 00:05:50,980 --> 00:05:53,770 Now we have points in the middle. 80 00:05:54,730 --> 00:05:58,720 And we have points in the, uh, in the boundary. 81 00:05:58,720 --> 00:06:08,200 So just to be visually, uh, we put it, we put it to green, we have these points, that is. 82 00:06:10,550 --> 00:06:11,570 Synthesize. 83 00:06:12,870 --> 00:06:14,280 We have these points. 84 00:06:14,280 --> 00:06:19,170 The domain, which will be located randomly spread in the middle. 85 00:06:20,150 --> 00:06:23,000 And of course, in the in the time domain. 86 00:06:23,000 --> 00:06:26,900 So we are calculating not just X and Y, we're also calculating in the time. 87 00:06:27,080 --> 00:06:30,500 And of course, we have the other points. 88 00:06:30,590 --> 00:06:31,880 We'll put them. 89 00:06:32,640 --> 00:06:33,390 Maybe. 90 00:06:34,450 --> 00:06:37,300 Uh, this like orange. 91 00:06:37,300 --> 00:06:41,830 We we will be generated points around here also. 92 00:06:44,450 --> 00:06:46,760 Randomly, but only on the boundary. 93 00:06:47,030 --> 00:06:47,930 So this. 94 00:06:47,930 --> 00:06:49,160 This too is. 95 00:06:49,190 --> 00:06:49,940 This is. 96 00:06:50,640 --> 00:06:52,500 The functions of this. 97 00:06:53,570 --> 00:06:54,080 Okay. 98 00:06:54,110 --> 00:06:54,790 So. 99 00:06:54,800 --> 00:06:56,690 So t initial. 100 00:06:58,430 --> 00:07:04,820 We have the time at the initial condition is torch dot. 101 00:07:05,860 --> 00:07:06,520 Zero. 102 00:07:07,450 --> 00:07:08,320 Like. 103 00:07:10,010 --> 00:07:10,610 Picks. 104 00:07:11,560 --> 00:07:12,900 It actually doesn't matter. 105 00:07:12,900 --> 00:07:19,560 We can just put it to evil because all of them have the same. 106 00:07:20,400 --> 00:07:21,030 Size. 107 00:07:22,440 --> 00:07:23,400 Number of points. 108 00:07:23,400 --> 00:07:23,850 Yeah. 109 00:07:23,880 --> 00:07:28,710 They all have a number of points and one column, so it doesn't really matter. 110 00:07:29,130 --> 00:07:29,880 Whatever it is. 111 00:07:29,880 --> 00:07:32,280 But because if we can call it. 112 00:07:32,490 --> 00:07:33,060 Yeah. 113 00:07:33,810 --> 00:07:38,160 T initial equals initial. 114 00:07:39,270 --> 00:07:45,060 Condition which the function we wrote and what we need to pass. 115 00:07:47,140 --> 00:07:47,920 X and Y. 116 00:07:49,350 --> 00:07:52,020 Y, X and Y because. 117 00:07:53,990 --> 00:07:55,490 We have to pass. 118 00:07:55,520 --> 00:07:59,600 We need the initial condition to be on the whole domain, not the boundary condition. 119 00:08:00,530 --> 00:08:02,030 These are just boundary points. 120 00:08:02,030 --> 00:08:06,500 So now we have the initial time, which is zero. 121 00:08:06,800 --> 00:08:13,670 We have a we take these X and Y these and we have the initial. 122 00:08:15,790 --> 00:08:21,130 Now the boundary value we talked about, there is a custom value in here, the boundary. 123 00:08:22,410 --> 00:08:23,460 A boundary. 124 00:08:26,180 --> 00:08:27,410 The costume. 125 00:08:27,470 --> 00:08:28,070 This one. 126 00:08:28,100 --> 00:08:29,240 The boundary condition. 127 00:08:29,240 --> 00:08:34,280 We have a custom value so we can set this custom value equals zero. 128 00:08:35,100 --> 00:08:35,880 And. 129 00:08:37,270 --> 00:08:39,490 Now we define the. 130 00:08:40,200 --> 00:08:41,880 Values of the boundary. 131 00:08:41,910 --> 00:08:42,720 You. 132 00:08:43,930 --> 00:08:44,590 A bond. 133 00:08:44,710 --> 00:08:46,510 Will, just let me just take it. 134 00:08:46,520 --> 00:08:48,280 I'm not the best in spelling. 135 00:08:49,140 --> 00:08:51,540 So be boundary. 136 00:08:52,850 --> 00:08:54,560 In the X direction. 137 00:08:54,560 --> 00:08:55,520 Equals. 138 00:08:55,700 --> 00:08:56,720 Equals what? 139 00:08:56,720 --> 00:08:57,890 The boundary condition. 140 00:08:58,010 --> 00:08:59,190 We need the whole thing. 141 00:08:59,210 --> 00:09:00,440 The boundary condition. 142 00:09:01,140 --> 00:09:02,310 The boundary condition. 143 00:09:02,310 --> 00:09:03,030 At what? 144 00:09:04,570 --> 00:09:05,920 We need to pass. 145 00:09:08,140 --> 00:09:08,440 The. 146 00:09:08,440 --> 00:09:08,950 What is it? 147 00:09:10,320 --> 00:09:12,480 No, this is the training, the boundary. 148 00:09:12,660 --> 00:09:21,420 This one we need X, Y and T, So we need the boundary X and the boundary Y and the T, of course, 149 00:09:21,450 --> 00:09:23,200 T, We don't. 150 00:09:23,400 --> 00:09:27,000 Just remember, we generated these points at random time. 151 00:09:27,750 --> 00:09:29,610 So here we take it. 152 00:09:29,880 --> 00:09:31,320 A boundary condition. 153 00:09:31,880 --> 00:09:33,060 And what is it? 154 00:09:33,920 --> 00:09:34,420 Yeah. 155 00:09:34,430 --> 00:09:35,240 This. 156 00:09:36,840 --> 00:09:37,440 Exp. 157 00:09:40,340 --> 00:09:41,930 We'll just take it like this. 158 00:09:42,000 --> 00:09:43,610 Wipe and. 159 00:09:44,580 --> 00:09:50,820 A teepee and we need to pass the custom value that is. 160 00:09:51,680 --> 00:09:53,150 These points. 161 00:09:53,630 --> 00:09:57,260 These points will have a custom value of. 162 00:09:59,370 --> 00:10:01,460 Uh, well, zero, basically. 163 00:10:01,530 --> 00:10:08,130 So the thing is, if you notice, like if we this initial condition, we only have x, Y, and this 164 00:10:08,130 --> 00:10:09,390 we have X, Y and Z. 165 00:10:09,420 --> 00:10:10,590 Why do we have this? 166 00:10:11,310 --> 00:10:15,900 Because in initial condition, it's always time is zero. 167 00:10:16,140 --> 00:10:21,540 In in in the boundary condition, time can change and or time will change. 168 00:10:21,540 --> 00:10:28,230 And we're actually sampling the whole time domain in different like random basically. 169 00:10:29,340 --> 00:10:32,010 So here we have the boundary condition. 170 00:10:32,100 --> 00:10:39,180 We define it in for the x like boundary in the X direction. 171 00:10:39,180 --> 00:10:42,870 And then we have the Y direction. 172 00:10:42,870 --> 00:10:46,620 And this time we will use these boundary points. 173 00:10:46,620 --> 00:10:47,190 But. 174 00:10:47,190 --> 00:10:49,530 But we need to flip them. 175 00:10:51,270 --> 00:10:52,200 Flipping them. 176 00:10:52,530 --> 00:10:56,820 The reason is, is is that I think we we don't Yeah. 177 00:10:56,820 --> 00:10:58,890 I think it's better to do like this. 178 00:10:59,280 --> 00:11:02,430 We have these boundaries that is going to be generated. 179 00:11:02,700 --> 00:11:04,640 We divide them on two. 180 00:11:04,650 --> 00:11:12,010 So we will do the boundary condition for both. 181 00:11:12,030 --> 00:11:13,320 Sorry, this is you. 182 00:11:15,920 --> 00:11:18,920 For the random points. 183 00:11:18,920 --> 00:11:25,310 And because we divide on two, it's better to have them two ways just because it's random. 184 00:11:25,310 --> 00:11:31,250 We just flip it to basically multiply the amount of points we have. 185 00:11:32,590 --> 00:11:34,210 So here. 186 00:11:34,690 --> 00:11:35,710 That's it. 187 00:11:35,830 --> 00:11:38,890 That's it for the optimization. 188 00:11:39,610 --> 00:11:45,520 Or we put the optimum and then we call these initial conditions and boundary conditions. 189 00:11:45,550 --> 00:11:49,590 The next way is we will need to calculate the losses. 190 00:11:49,600 --> 00:12:00,010 It's a little bit maybe confusing and to define the to make everything random rather than a mesh, but 191 00:12:00,010 --> 00:12:11,770 it actually uses the power of the neural networks by having random we don't care about mesh is is this 192 00:12:11,770 --> 00:12:17,560 is why I wrote the code like this because I want to show you decide where these points are. 193 00:12:17,590 --> 00:12:18,970 You decide. 194 00:12:20,660 --> 00:12:23,570 How they are spread and you don't really care to have a mesh.