1 00:00:00,400 --> 00:00:02,770 So now let's start building the data. 2 00:00:02,800 --> 00:00:10,720 So the first thing is, well, we we start by this the easiest thing. 3 00:00:10,720 --> 00:00:14,080 So let's first define the initial condition. 4 00:00:14,080 --> 00:00:16,570 So def. 5 00:00:17,470 --> 00:00:18,880 Initial. 6 00:00:19,930 --> 00:00:21,460 Condition can. 7 00:00:23,740 --> 00:00:24,640 Edition. 8 00:00:25,370 --> 00:00:31,190 Well, take X and Y and it will return the initial condition. 9 00:00:31,190 --> 00:00:33,710 So torch dot sine. 10 00:00:33,740 --> 00:00:41,120 This is of course we can, um, I mean we can add here like if we want a little bit, uh. 11 00:00:42,310 --> 00:00:43,810 More customization. 12 00:00:43,810 --> 00:00:45,310 But let's take it simple. 13 00:00:45,310 --> 00:00:53,200 And when just we return the actual initial condition as we write the equation in here, not kind of 14 00:00:53,200 --> 00:00:53,800 customized. 15 00:00:53,800 --> 00:00:58,930 If you want to customize it, you need to actually go to here and change the return function. 16 00:00:58,930 --> 00:01:02,800 So dot sign torch.py. 17 00:01:02,830 --> 00:01:09,250 Of course we can use math.py as well, but let's just use torches as it will work. 18 00:01:09,250 --> 00:01:11,170 And then torch. 19 00:01:11,950 --> 00:01:12,970 Dot sign. 20 00:01:14,090 --> 00:01:17,660 And torch dot pi. 21 00:01:18,940 --> 00:01:19,930 Multiply. 22 00:01:20,360 --> 00:01:21,610 Multiply two. 23 00:01:21,650 --> 00:01:22,240 Y. 24 00:01:23,020 --> 00:01:32,200 So this is the initial condition and it will give us a like a kind of hump, a peak around the zero. 25 00:01:32,200 --> 00:01:38,110 And as the the time goes, this peak will collapse. 26 00:01:38,140 --> 00:01:39,460 It will diffuse. 27 00:01:40,300 --> 00:01:46,410 You can use other, of course, networks or other sorry like functions. 28 00:01:46,450 --> 00:01:49,120 Def boundary. 29 00:01:49,160 --> 00:01:50,290 Boundary. 30 00:01:52,180 --> 00:01:54,130 Same condition can. 31 00:01:55,180 --> 00:01:57,130 On duration. 32 00:01:57,820 --> 00:02:05,590 We will have X, we will have Y, and we will have T, and we have a custom value. 33 00:02:06,220 --> 00:02:07,150 Custom. 34 00:02:08,200 --> 00:02:12,880 Value and this will return. 35 00:02:13,930 --> 00:02:15,280 Torch dot. 36 00:02:16,240 --> 00:02:17,200 Full. 37 00:02:17,200 --> 00:02:22,570 Like basically we will have the same. 38 00:02:24,510 --> 00:02:28,320 X the same shape as x. 39 00:02:29,080 --> 00:02:32,350 And with we will give it the custom value. 40 00:02:32,530 --> 00:02:33,880 Custom. 41 00:02:35,390 --> 00:02:43,130 And this core value, of course, I mean, in this example, it's a bit kind of simple. 42 00:02:43,130 --> 00:02:51,710 So x y it will be A3A box like a three, not triangle or a square. 43 00:02:51,830 --> 00:02:54,050 We have X and Y is the same. 44 00:02:54,050 --> 00:02:59,330 And so this shape is the same for. 45 00:02:59,330 --> 00:03:06,290 So if we if we have different like if we have X and Y is different, you need to return a different 46 00:03:06,290 --> 00:03:06,730 way. 47 00:03:06,740 --> 00:03:10,850 Of course this is it depends on you as a coder. 48 00:03:10,850 --> 00:03:14,840 And here I'm just trying to keep things as simple as possible. 49 00:03:14,840 --> 00:03:25,040 This is a box or the problem we are trying to to, to make or to solve is as simple as or it is just 50 00:03:25,130 --> 00:03:30,200 a triangle, a square with x and Y. 51 00:03:30,200 --> 00:03:31,460 And then. 52 00:03:33,930 --> 00:03:38,430 With X and y and x will equal to Y. 53 00:03:38,430 --> 00:03:47,430 And then what we will have is we will have something like in the middle, a hump like this, that we 54 00:03:47,430 --> 00:03:59,040 will give it the the value of this will be computed or it will be generated by the sine wave, the sine 55 00:03:59,040 --> 00:03:59,610 wave. 56 00:03:59,610 --> 00:04:08,700 And then as the time goes and of course, this this direction, this direction is you or the temperature. 57 00:04:08,700 --> 00:04:13,740 And as the time goes, the heat will start to diffuse. 58 00:04:13,950 --> 00:04:14,850 So. 59 00:04:15,660 --> 00:04:17,880 So this is what we what we're doing. 60 00:04:17,880 --> 00:04:21,990 And so initial condition, boundary condition, shift, enter. 61 00:04:22,500 --> 00:04:25,560 Now, after that, what do we need to do? 62 00:04:26,910 --> 00:04:29,550 So far, we don't have really a domain. 63 00:04:29,550 --> 00:04:35,310 We don't we just have a function that you give it X and Y, give you back the initial condition and 64 00:04:35,310 --> 00:04:37,890 x and Y give you back the Or. 65 00:04:37,890 --> 00:04:40,590 Yeah, X and y and give you back the boundary condition. 66 00:04:40,650 --> 00:04:48,450 So now what we want to do is we want to a function that generate a training data. 67 00:04:49,020 --> 00:04:52,740 Training data. 68 00:04:52,740 --> 00:04:58,590 And actually this is more like, well, the correct thing is, of course it's all training data. 69 00:04:58,590 --> 00:05:04,320 But the correct analogy here is this function is going to generate a domain points. 70 00:05:04,500 --> 00:05:18,690 So generating the points we start by x equals torch dot random and we will have the number of points 71 00:05:18,840 --> 00:05:26,650 and so that we will have the rows, it will be number of points, say 1000 point whatever number of 72 00:05:26,650 --> 00:05:28,330 points we want for training. 73 00:05:28,330 --> 00:05:30,700 And the column is one. 74 00:05:30,850 --> 00:05:34,600 So for our case, we don't need more than one. 75 00:05:34,600 --> 00:05:40,780 And actually this way of generation it will generate points from 0 to 1. 76 00:05:40,930 --> 00:05:46,480 Here requires grad equals. 77 00:05:46,480 --> 00:05:47,320 True. 78 00:05:48,280 --> 00:05:49,150 And. 79 00:05:49,670 --> 00:05:51,860 We do the same thing for. 80 00:05:52,880 --> 00:05:53,770 Why. 81 00:05:53,780 --> 00:05:55,400 And same thing for tea. 82 00:05:55,730 --> 00:05:58,670 So why random? 83 00:05:59,000 --> 00:06:00,350 All of it is the same. 84 00:06:00,590 --> 00:06:03,440 And tea also the same. 85 00:06:04,900 --> 00:06:11,190 And returns the X, Y and T. 86 00:06:12,610 --> 00:06:14,350 And we shift enter. 87 00:06:14,680 --> 00:06:20,830 So now we have the first function is initial condition, the second function boundary condition. 88 00:06:20,920 --> 00:06:23,830 Third function is the function that generates a point. 89 00:06:24,250 --> 00:06:30,200 The fourth thing is we need to do is all these are just points in the middle. 90 00:06:30,220 --> 00:06:33,370 We need another generator or this. 91 00:06:33,370 --> 00:06:40,030 These points are just spread all the way around the network, but around the domain. 92 00:06:40,030 --> 00:06:42,910 But what we need is we need a function. 93 00:06:44,460 --> 00:06:48,150 That generates points Generate boundary points. 94 00:06:49,170 --> 00:06:50,100 Rate. 95 00:06:50,580 --> 00:06:52,080 Boundary. 96 00:06:54,530 --> 00:06:55,250 Points. 97 00:06:55,250 --> 00:06:59,300 And we will have also the same thing, number of points. 98 00:06:59,660 --> 00:07:03,230 And then what we will do is X. 99 00:07:05,450 --> 00:07:11,600 Boundary equals torch dot. 100 00:07:12,800 --> 00:07:13,580 Tensor. 101 00:07:14,900 --> 00:07:20,780 And we will generate from zero 0 to 1.0. 102 00:07:22,710 --> 00:07:24,360 Dot re. 103 00:07:25,560 --> 00:07:26,280 Pete. 104 00:07:26,490 --> 00:07:30,780 This thing of number. 105 00:07:32,590 --> 00:07:33,580 Number. 106 00:07:35,060 --> 00:07:36,290 A number of points. 107 00:07:40,320 --> 00:07:41,370 Over to. 108 00:07:42,800 --> 00:07:43,280 To. 109 00:07:45,330 --> 00:07:46,260 And. 110 00:07:47,510 --> 00:07:48,350 Why? 111 00:07:51,780 --> 00:07:52,870 Boundary. 112 00:07:54,030 --> 00:07:57,720 Will equals torch dot tensor. 113 00:07:58,770 --> 00:07:59,640 And. 114 00:08:00,800 --> 00:08:01,960 Uh, will. 115 00:08:02,060 --> 00:08:02,750 Random. 116 00:08:04,080 --> 00:08:04,830 Rent. 117 00:08:05,730 --> 00:08:10,500 And we will also have them as number of points. 118 00:08:14,410 --> 00:08:20,490 One thing to note is this operator like divide. 119 00:08:20,500 --> 00:08:28,480 Divide by two means we need to repeat this operation, the tensor like generating having this tensor 120 00:08:28,480 --> 00:08:31,060 from 0 to 1 two. 121 00:08:31,570 --> 00:08:37,300 Well, basically, let's say if we have ten times, we need to repeat it five times. 122 00:08:37,420 --> 00:08:45,460 If the number of points enter is a little bit odd, like let's say like 11, the 11 divide by two, 123 00:08:45,490 --> 00:08:51,580 it will be five point something and then then it will be basically five again. 124 00:08:51,700 --> 00:08:58,990 So why boundary equals tensor dot random. 125 00:08:58,990 --> 00:09:08,260 And now we just generate so we make first the first tensor, it will generate a point that is like kind 126 00:09:08,260 --> 00:09:20,120 of a spaced like in a mesh wise and this one, it will generate just random values from a like just 127 00:09:20,150 --> 00:09:28,250 number of points that it will be random for Y, but here it will be only zero and one. 128 00:09:28,250 --> 00:09:38,720 So the tensor will be at zero and at one and it will keep having a so the x axis has to be left or right 129 00:09:38,720 --> 00:09:42,080 and the y axis we just random whatever it is. 130 00:09:43,600 --> 00:09:46,720 This is the boundary, of course, in the x axis. 131 00:09:46,720 --> 00:09:50,860 But what we need to do is we need also to generate boundaries in the y axis. 132 00:09:50,860 --> 00:10:01,240 For that, we need to if we make another random thing if torch dot random. 133 00:10:02,220 --> 00:10:03,150 More than. 134 00:10:03,840 --> 00:10:09,360 Well, like, basically it will generate from 0 to 1 random value or 0.5. 135 00:10:10,660 --> 00:10:12,460 And X. 136 00:10:13,460 --> 00:10:16,490 We just switched them Expounder. 137 00:10:18,440 --> 00:10:19,610 Why boundary. 138 00:10:22,170 --> 00:10:23,100 Equals. 139 00:10:23,130 --> 00:10:27,240 We switch the thing, so x boundary will be y boundary. 140 00:10:27,870 --> 00:10:30,330 And why boundary. 141 00:10:30,360 --> 00:10:32,010 This one will be expanded. 142 00:10:34,480 --> 00:10:35,710 Why did we do that? 143 00:10:36,280 --> 00:10:46,000 What we did is we went to the domain and from 0 to 1 we made some points or we want to make these points. 144 00:10:46,000 --> 00:10:52,780 That is X is set a value of zero and or and one. 145 00:10:52,780 --> 00:10:58,420 So x is is zero and y is x is zero. 146 00:10:58,420 --> 00:11:01,780 And the first value and the second value is one. 147 00:11:01,840 --> 00:11:05,950 And what we did is we generate random values around it. 148 00:11:05,950 --> 00:11:11,470 So here we have X will be zero and x will be one. 149 00:11:11,860 --> 00:11:19,870 And what we did, the first thing is we went here and we made just any random value in this location, 150 00:11:19,870 --> 00:11:23,830 just any random values here and here. 151 00:11:24,870 --> 00:11:27,510 The same thing here in random values. 152 00:11:27,510 --> 00:11:33,840 And then what we did is we took part of these random values and we put them here. 153 00:11:35,710 --> 00:11:41,120 So now we have boundary values around this area. 154 00:11:41,140 --> 00:11:46,030 So the last thing we need to do is return. 155 00:11:50,140 --> 00:11:52,930 X x boundary. 156 00:11:54,150 --> 00:11:56,070 And we reshape it. 157 00:11:57,840 --> 00:11:59,640 Into minus one. 158 00:11:59,640 --> 00:12:00,240 One. 159 00:12:01,010 --> 00:12:04,220 So one only one point. 160 00:12:04,580 --> 00:12:06,710 So the dimension is the rows. 161 00:12:06,740 --> 00:12:10,790 And it is just to make sure it is only one row. 162 00:12:11,810 --> 00:12:13,850 Same thing for Y. 163 00:12:21,310 --> 00:12:22,480 For why? 164 00:12:25,230 --> 00:12:27,600 And we reshape it this way. 165 00:12:29,950 --> 00:12:36,190 No, this is generating the boundary points. 166 00:12:36,220 --> 00:12:44,020 Now what we need is to generate the training data, the boundary training data generate. 167 00:12:45,190 --> 00:12:46,390 Boundary. 168 00:12:48,200 --> 00:12:49,430 Raining. 169 00:12:51,000 --> 00:12:56,100 Data and we will add number number of points. 170 00:12:59,230 --> 00:13:00,370 And here. 171 00:13:01,190 --> 00:13:07,640 What we will do is, Well, we need the first generator. 172 00:13:08,910 --> 00:13:17,220 Which is generating the points, this function and we will pass on the number of points. 173 00:13:18,920 --> 00:13:23,090 And think I need a little bit space. 174 00:13:23,810 --> 00:13:25,280 And then the. 175 00:13:27,390 --> 00:13:32,160 Time will be torch dot again random. 176 00:13:32,160 --> 00:13:34,470 We don't really care. 177 00:13:34,770 --> 00:13:38,160 Number of points. 178 00:13:39,680 --> 00:13:45,320 And we will have the column one and requires. 179 00:13:46,880 --> 00:13:47,750 Great. 180 00:13:49,720 --> 00:13:50,140 And. 181 00:13:52,210 --> 00:13:53,980 That equals true. 182 00:13:56,020 --> 00:13:56,980 Return. 183 00:13:58,690 --> 00:13:59,380 Expunging. 184 00:14:00,810 --> 00:14:01,860 Why boundary. 185 00:14:03,610 --> 00:14:04,180 Empty. 186 00:14:07,340 --> 00:14:08,030 And shift enter. 187 00:14:09,660 --> 00:14:10,740 With that. 188 00:14:11,010 --> 00:14:13,650 We'll just make housekeeping a little bit. 189 00:14:15,610 --> 00:14:16,390 Just like this. 190 00:14:16,840 --> 00:14:18,760 So with that, what did we do? 191 00:14:18,790 --> 00:14:20,800 We actually generated. 192 00:14:22,260 --> 00:14:24,300 A boundary point. 193 00:14:24,300 --> 00:14:28,350 And with this function, what we did, we ordered a boundary point. 194 00:14:28,350 --> 00:14:33,600 And this function gave us this boundary point that is split somehow, randomly. 195 00:14:33,600 --> 00:14:39,200 And the other one is we just took some random from 0 to 1. 196 00:14:39,220 --> 00:14:43,860 The time just we took again some random points in time. 197 00:14:44,460 --> 00:14:49,260 Again, we're training the network is not really explicit solution. 198 00:14:50,900 --> 00:14:52,610 It's more like implicit solution. 199 00:14:52,610 --> 00:14:56,300 But but this time we don't really have a grid. 200 00:14:56,330 --> 00:14:57,320 We don't need a grid. 201 00:14:57,350 --> 00:15:04,280 We just randomly separate these points or generate these points to actually get the required data.