1 00:00:01,150 --> 00:00:07,330 Now what we need to do is to calculate the losses. 2 00:00:07,360 --> 00:00:14,380 Now losses we have the residual loss or what we call loss or the equation loss. 3 00:00:14,620 --> 00:00:17,470 And the other loss we have is the data loss. 4 00:00:17,620 --> 00:00:21,280 So the first thing is let's calculate the residual loss. 5 00:00:21,280 --> 00:00:26,680 And the residual loss will basically depends on equation. 6 00:00:26,980 --> 00:00:32,980 So in order to do that, let's have a class or not a class, actually. 7 00:00:34,160 --> 00:00:36,170 Diff means like function. 8 00:00:38,720 --> 00:00:47,450 And what we will have, we will have X, we have Y, we have T, and we have the model. 9 00:00:48,350 --> 00:00:49,520 What do we need? 10 00:00:49,550 --> 00:00:52,480 Now this a pde. 11 00:00:53,180 --> 00:00:58,220 What it, what it going to to do is we need to calculate the. 12 00:00:59,040 --> 00:01:04,020 The the basically the residual that is, um. 13 00:01:07,470 --> 00:01:08,450 I will explain it. 14 00:01:08,460 --> 00:01:14,370 So basically, you have this part of the equation, all of this part of the equation. 15 00:01:14,370 --> 00:01:24,000 And then by moving this part to here, we will get an equation that will basically look like a like 16 00:01:24,000 --> 00:01:25,080 this. 17 00:01:27,310 --> 00:01:28,450 Minus. 18 00:01:31,070 --> 00:01:33,620 I will move the things here. 19 00:01:34,820 --> 00:01:37,370 Minus due. 20 00:01:39,020 --> 00:01:44,180 RT and all of this will will equal to. 21 00:01:46,050 --> 00:01:46,710 Zero. 22 00:01:48,670 --> 00:01:49,450 Now. 23 00:01:50,170 --> 00:01:52,570 The reality is it will not equal to zero. 24 00:01:53,350 --> 00:01:54,640 And this. 25 00:01:56,750 --> 00:01:59,690 This will have or not this one. 26 00:01:59,720 --> 00:02:02,150 This will have a value. 27 00:02:02,540 --> 00:02:09,020 So what we will do, we'll try to the loss will be we will have this equation that equals to zero. 28 00:02:09,050 --> 00:02:10,820 Of course, it can work for any equation. 29 00:02:10,820 --> 00:02:14,300 Just put everything on one side and equal it to zero. 30 00:02:14,390 --> 00:02:16,520 But make sure later we will see. 31 00:02:16,520 --> 00:02:19,160 Like this zero means every data point. 32 00:02:19,160 --> 00:02:21,710 So we need to be careful about the size. 33 00:02:21,980 --> 00:02:23,580 I will show how to do this. 34 00:02:23,600 --> 00:02:27,080 So this is has to be zero. 35 00:02:27,230 --> 00:02:30,800 If we calculate how much is this, then we will get a value. 36 00:02:30,800 --> 00:02:34,400 And this value we will say, I want the loss to be zero. 37 00:02:38,090 --> 00:02:39,140 So let's do it. 38 00:02:39,410 --> 00:02:41,960 In order to do it first, we need to. 39 00:02:42,440 --> 00:02:44,240 Well, basically, we will. 40 00:02:45,650 --> 00:02:46,580 Start. 41 00:02:47,130 --> 00:02:48,260 Just a minute. 42 00:02:52,190 --> 00:02:56,060 Yeah, we will start by the input data. 43 00:02:56,630 --> 00:02:58,430 Input data. 44 00:02:59,220 --> 00:03:02,700 Equals torch dot. 45 00:03:06,400 --> 00:03:14,350 Concatenate X, y, and T with a dimension equals one. 46 00:03:15,660 --> 00:03:21,420 The main reason why we said like this because X, Y, and T is actually a tensor here. 47 00:03:21,420 --> 00:03:27,990 We generated a tensor like we remember here a random tensor of x, a random tensor of Y and a random 48 00:03:28,020 --> 00:03:29,310 tensor with t. 49 00:03:29,340 --> 00:03:39,540 Now we will cut concatenate these x, y and T because we want to pass this these values to the model 50 00:03:39,660 --> 00:03:42,510 and basically get the. 51 00:03:43,210 --> 00:03:44,560 Input data. 52 00:03:45,580 --> 00:03:47,620 Look, we get that you. 53 00:03:48,480 --> 00:03:53,160 So the model will, will, will generate the data points. 54 00:03:53,160 --> 00:03:56,910 So we have three tensors we use them to make. 55 00:03:57,030 --> 00:03:58,500 Well, basically data. 56 00:03:58,710 --> 00:04:06,120 And then this data we put them into the model and by that we generate a values of U. 57 00:04:06,660 --> 00:04:09,750 After that, what we need to do is we need to. 58 00:04:10,800 --> 00:04:12,330 And differentiate you. 59 00:04:12,330 --> 00:04:14,760 And this is another. 60 00:04:16,120 --> 00:04:19,210 Like another way to do it, but okay, I will write it this way. 61 00:04:19,510 --> 00:04:29,350 X UX and UI equals torch dot the same we did like before auto. 62 00:04:30,120 --> 00:04:30,930 Grad. 63 00:04:32,040 --> 00:04:39,750 The things we want to differentiate is you risk in corresponding to x, y. 64 00:04:39,780 --> 00:04:45,570 Now here x, Y, or we just write two times like x here and y here. 65 00:04:48,860 --> 00:04:50,450 And actually I'll just put it. 66 00:04:50,490 --> 00:04:53,420 I like the simplification. 67 00:04:54,510 --> 00:05:04,030 Just like this you and then we want to like, basically two x. 68 00:05:05,460 --> 00:05:06,680 And this too. 69 00:05:06,690 --> 00:05:07,410 Why? 70 00:05:11,250 --> 00:05:12,000 Grad. 71 00:05:13,210 --> 00:05:14,770 Outputs. 72 00:05:15,820 --> 00:05:16,780 Equals. 73 00:05:16,900 --> 00:05:18,940 Same thing we did before. 74 00:05:22,240 --> 00:05:23,290 George. 75 00:05:25,320 --> 00:05:25,780 Dot. 76 00:05:26,950 --> 00:05:29,020 Ones like. 77 00:05:29,750 --> 00:05:30,470 You. 78 00:05:31,720 --> 00:05:32,860 Create. 79 00:05:34,210 --> 00:05:35,230 Graph. 80 00:05:37,750 --> 00:05:38,380 Equals. 81 00:05:38,380 --> 00:05:39,160 True. 82 00:05:40,750 --> 00:05:41,590 She threw. 83 00:05:43,770 --> 00:05:45,360 And retain. 84 00:05:46,700 --> 00:05:47,600 Graph. 85 00:05:48,490 --> 00:05:50,020 Equals also true. 86 00:05:51,840 --> 00:05:54,210 This one we repeated. 87 00:05:55,740 --> 00:05:56,340 It. 88 00:05:58,030 --> 00:05:58,780 Two times. 89 00:05:58,780 --> 00:06:08,830 Of course we can do it like this or just just put it this way or just make it a bit concise. 90 00:06:09,470 --> 00:06:10,440 UX. 91 00:06:11,730 --> 00:06:17,430 You I am just worried about might be another value that will pop up here. 92 00:06:17,430 --> 00:06:21,420 And yeah, need to be careful about the dimension. 93 00:06:22,530 --> 00:06:24,660 So this. 94 00:06:28,100 --> 00:06:29,510 This is the first one. 95 00:06:30,190 --> 00:06:31,930 And what we need to use. 96 00:06:31,930 --> 00:06:33,980 Actually, we're not really using this one. 97 00:06:34,000 --> 00:06:37,090 What we will do is use x x. 98 00:06:38,730 --> 00:06:45,270 The second derivative we will is the same, but we need to be attention of some things. 99 00:06:45,840 --> 00:06:47,820 We need to use. 100 00:06:49,320 --> 00:06:49,980 To. 101 00:06:52,560 --> 00:06:56,640 To to use for differentiation. 102 00:06:57,720 --> 00:06:59,280 This is the second derivative. 103 00:06:59,280 --> 00:07:02,280 And here it will be. 104 00:07:02,790 --> 00:07:04,230 For X. 105 00:07:05,130 --> 00:07:08,790 Tortured, cried like not you. 106 00:07:08,820 --> 00:07:14,640 It will be UX and everything will be the same. 107 00:07:14,640 --> 00:07:15,480 But. 108 00:07:16,240 --> 00:07:16,870 Here. 109 00:07:16,870 --> 00:07:18,640 We need only zero. 110 00:07:19,940 --> 00:07:24,590 And we said about this, it will well, it will make a tuple and we don't want the tuple. 111 00:07:24,590 --> 00:07:25,970 We want just the value. 112 00:07:25,970 --> 00:07:28,880 It would put one value, one tensor value in the tuple. 113 00:07:28,880 --> 00:07:30,830 And this is how it works. 114 00:07:30,830 --> 00:07:37,820 Like it's just this is of course, using PyTorch is not made only for pens, but we don't need the tuple 115 00:07:37,820 --> 00:07:46,460 and we this is why we take this value of zero you and now we do the same thing with you. 116 00:07:46,940 --> 00:07:47,940 Well, we. 117 00:07:49,190 --> 00:07:52,070 You y y. 118 00:07:52,160 --> 00:07:54,950 And we also be careful here. 119 00:07:54,950 --> 00:08:05,960 You have to put UI and we have to derivate in relation to U to Y and here we have to pass y and graph, 120 00:08:06,140 --> 00:08:14,930 create, create, not like create here and then here, create, retrain, retain the graph equals true. 121 00:08:14,930 --> 00:08:17,840 Everything is seems in place. 122 00:08:17,840 --> 00:08:23,340 The only thing we need to care about is u t. 123 00:08:25,170 --> 00:08:26,190 Which is. 124 00:08:27,430 --> 00:08:28,680 The division. 125 00:08:28,680 --> 00:08:32,310 With time you will be with. 126 00:08:32,310 --> 00:08:33,810 You should be. 127 00:08:34,260 --> 00:08:36,000 So we. 128 00:08:37,080 --> 00:08:41,490 But you and we derive it with time. 129 00:08:43,070 --> 00:08:45,560 And we put again. 130 00:08:47,130 --> 00:08:48,630 You hear? 131 00:08:49,600 --> 00:08:50,710 And create graph. 132 00:08:50,710 --> 00:08:54,430 Everything is seems okay. 133 00:08:55,720 --> 00:08:58,450 Heat equation. 134 00:08:58,990 --> 00:09:00,430 Residual. 135 00:09:00,610 --> 00:09:01,720 Residual. 136 00:09:02,570 --> 00:09:06,200 Equals, I will put the coefficient of one. 137 00:09:07,310 --> 00:09:10,580 So we put it this is going to be the equation. 138 00:09:11,390 --> 00:09:14,330 Well, we have this will be one. 139 00:09:14,330 --> 00:09:16,880 So this will be one multiplied. 140 00:09:17,000 --> 00:09:22,700 The second derivative of u over x and the second derivative of u over y minus du over dt. 141 00:09:23,030 --> 00:09:27,130 So this is one multiplied UX x. 142 00:09:28,320 --> 00:09:30,360 Plus one multiplied. 143 00:09:31,530 --> 00:09:33,570 We'll just make it look nice. 144 00:09:33,730 --> 00:09:36,450 You y minus. 145 00:09:37,540 --> 00:09:38,140 UT. 146 00:09:40,820 --> 00:09:43,340 And we return. 147 00:09:44,310 --> 00:09:46,530 Of course, we just can't put it all of this here. 148 00:09:46,530 --> 00:09:49,260 But anyway, return residual. 149 00:09:52,170 --> 00:09:53,610 And shift enter. 150 00:09:55,220 --> 00:10:05,060 So we go back to the main loop and what we will do is we are going to define residual, which is the 151 00:10:05,060 --> 00:10:07,190 residual error equals. 152 00:10:09,070 --> 00:10:13,690 And we just pass and model. 153 00:10:18,080 --> 00:10:19,850 The model is here. 154 00:10:19,880 --> 00:10:25,010 Of course, the model will be we will will define it later in the actual training process. 155 00:10:25,010 --> 00:10:30,380 But basically the model in the in the training class, we have to pass it. 156 00:10:30,920 --> 00:10:33,350 So now we have the residual. 157 00:10:37,150 --> 00:10:38,320 Calculate the loss. 158 00:10:38,650 --> 00:10:43,000 Loss is in mean squared error. 159 00:10:43,030 --> 00:10:44,470 Keep it simple loss. 160 00:10:44,500 --> 00:10:47,650 Of course we will keep it simple. 161 00:10:47,680 --> 00:10:50,320 But in usual case it might. 162 00:10:50,350 --> 00:10:51,610 We need more maybe. 163 00:10:51,610 --> 00:10:53,800 Or different losses. 164 00:10:53,800 --> 00:10:55,270 But this is just. 165 00:10:55,270 --> 00:10:59,290 It works in this simple equations. 166 00:10:59,740 --> 00:11:02,590 Model dot torch Dot. 167 00:11:04,460 --> 00:11:05,810 Concatenate. 168 00:11:06,900 --> 00:11:07,550 Cut. 169 00:11:08,480 --> 00:11:11,180 And we will add X. 170 00:11:11,930 --> 00:11:15,980 Y and T initial. 171 00:11:18,910 --> 00:11:19,660 And. 172 00:11:20,520 --> 00:11:25,740 Uh, we need to consider the same dimension here. 173 00:11:26,250 --> 00:11:28,650 Dimension equals one. 174 00:11:29,820 --> 00:11:31,140 And. 175 00:11:32,860 --> 00:11:33,780 Plus. 176 00:11:38,310 --> 00:11:39,810 Or just. 177 00:11:47,960 --> 00:11:49,730 It's not shifting. 178 00:11:51,010 --> 00:11:51,820 Anyway. 179 00:11:51,820 --> 00:11:56,860 So we have we need again this loss. 180 00:12:01,990 --> 00:12:02,230 Mm. 181 00:12:17,030 --> 00:12:23,540 I wanted to be in this direction, but somehow it's not moving anyway, you initial. 182 00:12:24,290 --> 00:12:30,530 This initial value has to equal the points. 183 00:12:31,510 --> 00:12:36,250 That we calculated will equal the the output of the model itself. 184 00:12:37,060 --> 00:12:38,080 And. 185 00:12:44,770 --> 00:12:45,190 Yeah. 186 00:12:46,800 --> 00:12:47,910 And. 187 00:12:48,840 --> 00:12:52,350 The loss of you boundary. 188 00:12:55,740 --> 00:12:58,160 Will do the same. 189 00:12:58,170 --> 00:13:00,030 X underscore b. 190 00:13:01,100 --> 00:13:01,980 Y. 191 00:13:02,420 --> 00:13:02,810 Aw. 192 00:13:02,930 --> 00:13:03,170 Yeah. 193 00:13:03,260 --> 00:13:06,320 Y underscore V and this one. 194 00:13:07,310 --> 00:13:07,880 And. 195 00:13:14,070 --> 00:13:18,330 Yeah, not initial because we don't really need that. 196 00:13:19,170 --> 00:13:20,460 Because the time will change. 197 00:13:20,460 --> 00:13:23,130 Here is already it's only zero. 198 00:13:25,640 --> 00:13:26,540 And. 199 00:13:35,930 --> 00:13:41,180 And we have also the other loss we have. 200 00:13:43,970 --> 00:13:44,410 Now. 201 00:13:44,410 --> 00:13:45,790 It's kind of working. 202 00:14:00,380 --> 00:14:04,790 Is boundary condition of you. 203 00:14:05,030 --> 00:14:06,740 And here we flipped it. 204 00:14:06,740 --> 00:14:08,630 So why be? 205 00:14:09,570 --> 00:14:13,560 We flipped it to generate more points and dimension is the same. 206 00:14:14,550 --> 00:14:15,720 And. 207 00:14:18,260 --> 00:14:23,990 Here we have the last error is the residual error. 208 00:14:26,640 --> 00:14:27,900 So residual. 209 00:14:35,520 --> 00:14:40,620 And torch dot zero like. 210 00:14:41,960 --> 00:14:43,640 A residual. 211 00:14:43,730 --> 00:14:47,300 So it has to be the same size as, as I said. 212 00:14:54,270 --> 00:15:00,720 Now, after that, what we will do is loss dot back. 213 00:15:02,890 --> 00:15:05,080 Propagation backward. 214 00:15:06,450 --> 00:15:08,040 And the Optimizer. 215 00:15:08,170 --> 00:15:09,570 Optimizer. 216 00:15:13,080 --> 00:15:14,880 We will have a step. 217 00:15:18,990 --> 00:15:22,050 If it ration. 218 00:15:23,480 --> 00:15:27,870 Residual 100 equals or equals equal zero. 219 00:15:27,890 --> 00:15:33,740 Basically, we need to print it every 100 step print. 220 00:15:36,840 --> 00:15:38,430 Iteration. 221 00:15:39,840 --> 00:15:40,010 He. 222 00:15:41,460 --> 00:15:42,900 The iteration. 223 00:15:43,880 --> 00:15:45,410 And Los. 224 00:15:51,180 --> 00:15:52,110 Los. 225 00:15:57,520 --> 00:15:58,510 Los. 226 00:16:00,070 --> 00:16:00,760 This one. 227 00:16:02,660 --> 00:16:08,600 Okay, so this way I think it's quite good. 228 00:16:10,650 --> 00:16:16,260 Next step, of course, shift enter and we have a little bit problem. 229 00:16:21,110 --> 00:16:21,710 And. 230 00:16:25,980 --> 00:16:26,600 Are here. 231 00:16:26,610 --> 00:16:28,050 Extra one. 232 00:16:29,310 --> 00:16:31,050 Just to be safe. 233 00:16:31,050 --> 00:16:34,140 I think here it's better to have it like this. 234 00:16:35,730 --> 00:16:38,340 Like this will be continuation for this. 235 00:16:45,640 --> 00:16:48,580 And shift into seems okay. 236 00:16:49,270 --> 00:16:49,720 Okay. 237 00:16:49,750 --> 00:16:53,980 This now we already finished how to define the loss. 238 00:16:53,980 --> 00:16:56,410 And next step we will train this model.