[User] Please transcribe this text

[Assistant] And your task is to bounce the ball past the other player, uh, controlled by AI.


[User] Please transcribe this text

[Assistant] The human being would immediately, they may not win the game, but they would immediately understand the game. And would be able to successfully play it well enough to pretty quickly learn to beat the game.


[User] Please transcribe this text

[Assistant] They need to have a concept of control. What it means to control a paddle. They need to have a concept of a paddle. They need to have a concept of moving up and down.


[User] Please transcribe this text

[Assistant] And a ball and bouncing, they have to know, they have to have a, at least a loose concept of real world physics, that they can then project that real world physics onto the two dimensional world. All of these concepts are, uh, are concepts that you come to the table with. That's knowledge.


[User] Please transcribe this text

[Assistant] And the kind of way you transfer that knowledge from, uh, from your previous experience, from childhood to now, when you come to this game,


[User] Please transcribe this text

[Assistant] That is something is called reasoning. Whatever reasoning means.


[User] Please transcribe this text

[Assistant] And the question is whether through this same kind of process, you can see the entire world as a game of Pong. And reasoning is simply ability to simulate that game in your mind.


[User] Please transcribe this text

[Assistant] And learn very efficiently, much more efficiently than 200,000 iterations.


[User] Please transcribe this text

[Assistant] The other challenge of deep neural networks and machine learning broadly is you need big data and efficient learners as I said.


[User] Please transcribe this text

[Assistant] That data also needs to be supervised data. You need to have ground truth. Which is very costly for, so annotation.


[User] Please transcribe this text

[Assistant] A human being looking at a particular image, for example, and labeling that as something as, as a cat or a dog, whatever object is in the image, that's very costly.


[User] Please transcribe this text

[Assistant] And for particularly for, uh, neural networks, there's a lot of parameters to tune. There's a lot of hyperparameters. You need to figure out,


[User] Please transcribe this text

[Assistant] The network structure first. How does this network look? How many layers? How many hidden nodes?


[User] Please transcribe this text

[Assistant] What type of, uh, what type of activation function on each node? There's a lot of hyperparameters there. And then once you've built your network,


[User] Please transcribe this text

[Assistant] There's parameters for how you teach that network. There's learning rate, loss function, mini batch size, number of training iterations, uh, gradient update smoothing.


[User] Please transcribe this text

[Assistant] And the, selecting even the, uh, optimizer with which you, uh, with which you solve the various differential equations involved.


[User] Please transcribe this text

[Assistant] It's a topic of many research papers, certainly. It's rich enough for research papers, but it's also really challenging.


[User] Please transcribe this text

[Assistant] Loss function, or in the case of pong or games, a good...


[User] Please transcribe this text

[Assistant] I'm teaching a, uh, a network to play the game of Coast Runners. And the goal of Coast Runners...


[User] Please transcribe this text

[Assistant] Uh, and successfully complete a race against other people you're racing against.


[User] Please transcribe this text

[Assistant] Now, this network is an optimal one.


[User] Please transcribe this text

[Assistant] And what is figured out, that actually in the game,


[User] Please transcribe this text

[Assistant] It gets a lot of points for collecting certain objects along the path. So what you see is, it's figured out to go in a circle and collect those, those green, uh, turbo things.


[User] Please transcribe this text

[Assistant] And despite being on fire and hitting the wall and going through this whole process, it's actually achieved at least a local optima, given the reward function of maximizing the number of points. And so,


[User] Please transcribe this text

[Assistant] The, it's, it's figured out a way to earn a higher award while ignoring the implied bigger picture goal of finishing the race, which us as humans, uh, understand much better.


[User] Please transcribe this text

[Assistant] This raises for self-driving cars ethical questions.


[User] Please transcribe this text

[Assistant] It's hard to teach. It's hard to encode.


[User] Please transcribe this text

[Assistant] Formally define a utility function under which an intelligence system needs to operate. And that's made obvious even in a simple game. And so what, what is, yep, question.


[User] Please transcribe this text

[Assistant] So the question was, what's an example of a local optimum that, uh, an autonomous car, so similar to the coast racer, what would be the example in the real world for an autonomous vehicle? And it's a touchy subject, but it would certainly have to be involved.


[User] Please transcribe this text

[Assistant] The choices we make under near crashes and crashes. The choices a car makes when to avoid, uh, for example if the, if there's a crash imminent and there's no way you can stop.


[User] Please transcribe this text

[Assistant] Just prevent the crash. Do you, do you keep the driver safe or do you keep the other, uh, people safe? And there has to be some


[User] Please transcribe this text

[Assistant] Even if you, even if you don't choose to acknowledge it, even if it's only in the data and the learning that you do, there's an implied reward function there.


[User] Please transcribe this text

[Assistant] And we need to be aware of where that award function is, because it may find something. Until you actually see it, we won't know it. Once we see it, we'll realize that, oh,


[User] Please transcribe this text

[Assistant] That was a bad design. And that's the scary thing. It's hard to know ahead of time what that is.


[User] Please transcribe this text

[Assistant] So the recent breakthroughs from deep learning came,


[User] Please transcribe this text

[Assistant] Moore's law. CPUs are getting faster. A hundred times faster every decade. Then there's GPUs.


[User] Please transcribe this text

[Assistant] Also, the ability to train neural networks and GPUs and now ASICs has created a lot of, uh,


[User] Please transcribe this text

[Assistant] Well, first of all, in the 21st century, there's digitized data. There's larger data sets of digital data.


[User] Please transcribe this text

[Assistant] And now there is, that data, uh, is becoming more organized. Not just vaguely, uh, available data out there on the internet. It's actual organized data sets like ImageNet, uh, uh, certainly for natural language there's large data sets.


[User] Please transcribe this text

[Assistant] There is the algorithm innovations. Backprop, backpropagation, convolutional neural networks, LSTMs, all these different architectures for dealing with specific, uh, specific types of domains and tasks. There's the huge one.


[User] Please transcribe this text

[Assistant] There's infrastructure. There's, uh, on the software and the hardware side. There's Git, ability to share in an open source way, software.


[User] Please transcribe this text

[Assistant] There is pieces of software that, uh,


[User] Please transcribe this text

[Assistant] But, uh, that make robotics and make machine learning easier. Ross, TensorFlow. There's an Amazon Mechanical Turk.


[User] Please transcribe this text

[Assistant] Which allows for efficient, cheap annotation of large-scale datasets.


[User] Please transcribe this text

[Assistant] It's AWS in the cloud hosting, uh, machine learning, hosting the data and the compute.


[User] Please transcribe this text

[Assistant] And then there's a financial backing of large companies. Google, Facebook, Amazon. But really,


[User] Please transcribe this text

[Assistant] Nothing has changed. There really has not been any significant breakthroughs. We're using the ex- convolutional neural networks have been around since the 90s. Neural networks have been around since the 60s. There's been a few improvements.


[User] Please transcribe this text

[Assistant] But the hope is, that's in terms of methodology. The compute has really been the workhorse.


[User] Please transcribe this text

[Assistant] The ability to do, uh, the hundred fold improvement every decade.


[User] Please transcribe this text

[Assistant] Holds promise. And the question is whether that reasoning thing I talked about


[User] Please transcribe this text

[Assistant] Is all you need is a larger network? That is the open question.


[User] Please transcribe this text

[Assistant] So some terms for deep learning. First of all, deep learning is a PR term for neural networks.


[User] Please transcribe this text

[Assistant] For deep neural networks, for neural networks that have many layers.


[User] Please transcribe this text

[Assistant] It is symbolic term for the newly gained capabilities that compute has brought us. That, that training on GPUs have brought us.


[User] Please transcribe this text

[Assistant] So deep learning is a subset of machine learning. There's many other methods that are still effective.


[User] Please transcribe this text

[Assistant] First of all, multilayer perceptron, deep neural networks, recurrent neural networks, LSTM, long short term memory networks, CNN or CONVNETs, convolutional neural networks, deep belief networks, and the operations that will come up is convolution, pooling, activation functions, and back propagation.


[User] Please transcribe this text

[Assistant] So the question was, what is the purpose of the different layers in a neural network? What does it mean to have one configuration versus another? So a neural network is having several layers.


[User] Please transcribe this text

[Assistant] It's the only thing you have an understanding of is the inputs and the outputs.


[User] Please transcribe this text

[Assistant] You don't have a good understanding about what each layer does. There are mysterious things in neural networks.


[User] Please transcribe this text

[Assistant] So I'll talk about how with every layer it forms a higher level


[User] Please transcribe this text

[Assistant] A higher order representation of the input. So it's not like, uh, the first layer does localization, the second layer does path planning, the third layer, uh, does navigation, how you get from here to Florida. Or maybe it does. But we don't know.


[User] Please transcribe this text

[Assistant] For simple tasks, like for ImageNet, classifying cats versus dogs. We can tell what is the thing that the first layer does, the second layer, the third layer, and we'll look at that. But for driving, when you, as the input provides just the images and the output, the steering, it's still unclear what you learned.


[User] Please transcribe this text

[Assistant] Partially because we don't have neural networks that drive successfully yet.


[User] Please transcribe this text

[Assistant] So this, uh, this is, uh, the question was, do you, uh, do you, uh, does a neural network generate layers over time? Like does it grow?


[User] Please transcribe this text

[Assistant] A neural network is predefined, the architectures, the number of nodes, number of layers. That's all fixed. Unlike the human brain where neurons die and are born all the time. Neural network is pre-specified. That's it. That's all you get. And if you want to change that, you have to change that and then retrain everything. So it's fixed.


[User] Please transcribe this text

[Assistant] So what I encourage you is to proceed with caution because there's this feeling when you first teach a network with very little effort.


[User] Please transcribe this text

[Assistant] How to do some amazing tasks, like classify a face.


[User] Please transcribe this text

[Assistant] Uh, versus non-face, or your face versus other faces, or cats versus dogs. It's an incredible feeling.


[User] Please transcribe this text

[Assistant] And then you, there's definitely this feeling that I'm an expert. But what you realize is...


[User] Please transcribe this text

[Assistant] It, it's, you don't actually understand how it works.


[User] Please transcribe this text

[Assistant] And getting it to perform well for more generalized tasks, for larger scale datasets, for more useful applications, requires a lot of hyperparameter tuning. Figuring out how to tweak little things here and there, and still in the end, you don't understand why it works so damn well.


[User] Please transcribe this text

[Assistant] These deep neural network architectures is representation learning.


[User] Please transcribe this text

[Assistant] This is the difference between, uh, traditional machine learning methods, where


[User] Please transcribe this text

[Assistant] Having an image here is the input. The input to the network here is on the bottom. The output is at the top.


[User] Please transcribe this text

[Assistant] So, and the input is a single image of a person in this case. And so, the input specifically is all of the pixels in that image, RGB, the different colors of the pixels in the image. And over time,


[User] Please transcribe this text

[Assistant] What, what a network does is build a multi-resolutional representation of this data. The first layer,


[User] Please transcribe this text

[Assistant] Builds, uh, learns the concept of edges, for example.


[User] Please transcribe this text

[Assistant] The second layer starts to learn composition of those edges, corners, contours.


[User] Please transcribe this text

[Assistant] Then it starts to learn about object parts.


[User] Please transcribe this text

[Assistant] And finally, actually provide a label for the entities that are in the input.


[User] Please transcribe this text

[Assistant] And this is the difference between traditional machine learning methods, where the,


[User] Please transcribe this text

[Assistant] Concepts like edges and corners and contours are manually pre-specified by human.


[User] Please transcribe this text

[Assistant] By human beings, human experts for the particular domain.


[User] Please transcribe this text

[Assistant] And representation matters. Because...


[User] Please transcribe this text

[Assistant] For the Cartesian coordinates of this particular data set, where you want to design a machine learning system that tells the difference between green triangles and blue circles, is difficult. There's no line that separates them cleanly.


[User] Please transcribe this text

[Assistant] And if you were to ask a human being, a human expert in the field to try to draw that line, they would probably do a PhD on it and still not succeed. But a neural network can automatically figure out to remap that input.


[User] Please transcribe this text

[Assistant] Into polar coordinates, where the representation is such that it's an easily linearly separable data set.


[User] Please transcribe this text

[Assistant] And so, deep learning is a subset of representation learning, is a subset of machine learning, and a key subset of artificial intelligence.


[User] Please transcribe this text

[Assistant] Because of its ability to compute an arbitrary number of features at the core of the representation. So you're not, if you're trying to detect a cat in an image.


[User] Please transcribe this text

[Assistant] You're not specifying 215 specific features of cat ears and whiskers and so on that a human expert would specify.


[User] Please transcribe this text

[Assistant] You allow a neural network to discover tens of thousands of such features.


[User] Please transcribe this text

[Assistant] Which, maybe for cats you are an expert, but for a lot of objects you may never be able to sufficiently provide the


[User] Please transcribe this text

[Assistant] The key, the features which successfully would be used for identifying the object.


[User] Please transcribe this text

[Assistant] And so this kind of representation learning, one is easy in the sense that all you have to provide is inputs and outputs. All you need to provide is a data set that you care about without hand engineering features. And two, because of the


[User] Please transcribe this text

[Assistant] Uh, its ability to construct arbitrarily sized representations.


[User] Please transcribe this text

[Assistant] Deep neural networks are hungry for data. The more data we give them,


[User] Please transcribe this text

[Assistant] The more they're able to learn about this particular dataset.


[User] Please transcribe this text

[Assistant] So let's look at some applications.


[User] Please transcribe this text

[Assistant] Some cool things that deep neural networks have been able to accomplish up to this point. Let me go through them. First, the basic one.


[User] Please transcribe this text

[Assistant] It's a dataset, and it's a competition of classification and localization.


[User] Please transcribe this text

[Assistant] Where the task is given an image, identify what are the five most likely things in that image and what is the most likely and you have to do so correctly. So on the right there's an image of a leopard and you have to correctly classify that that is in fact a leopard. So they're able to do this pretty well. Given a specific image, determine that it's a leopard and


[User] Please transcribe this text

[Assistant] We started what's shown here on the x-axis is years, on the y-axis is error in classification. So starting from 2012 on the left with AlexNet, and today,


[User] Please transcribe this text

[Assistant] The error is decreased from 16%.


[User] Please transcribe this text

[Assistant] This picture of a leopard, there's a four, four percent, for four percent of those pictures of leopards, you would not say it's a leopard. That's human level performance. So for the first time in 2015, uh, convolutional neural networks outperform human beings. That in itself is incredible. That's something that seemed impossible. And now is, because it's done, is not as impressive.


[User] Please transcribe this text

[Assistant] But I just want to get to why that's so impressive. Because computer vision is hard.


[User] Please transcribe this text

[Assistant] We as human beings have evolved visual perception over millions of years, hundreds of millions of years.


[User] Please transcribe this text

[Assistant] So we take it for granted. But computer vision is really hard. Visual perception is really hard. There is illumination variability. So it's the same object. The only way we tell anything is from the shade, the reflection of light from that surface.


[User] Please transcribe this text

[Assistant] It's, it can be the same object with drastically, in terms of pixels, drastically different looking shapes. And we still know it's the same object. There is pose variability and occlusions.


[User] Please transcribe this text

[Assistant] Probably my favorite caption for an image, uh, for a figure in an academic paper is, deformable and truncated cat.


[User] Please transcribe this text

[Assistant] These are pictures, you know, cats are famously, uh,


[User] Please transcribe this text

[Assistant] They are deformable. They can take a lot of different shapes.


[User] Please transcribe this text

[Assistant] Uh, it's, it's arbitrary poses are, are, uh,


[User] Please transcribe this text

[Assistant] We still know it's a cat even when parts of it are not visible. And sometimes large parts of it are not visible. And then there's all the intraclass variability.


[User] Please transcribe this text

[Assistant] In interclass, all of these on the top two rows are cats. Many of them look drastically different. And the top bottom two rows are dogs. Also look drastically different.


[User] Please transcribe this text

[Assistant] And yet, some of the dogs look like cats, some of the cats look like dogs, and us human beings are pretty good at telling the difference, and we want computer vision to do better than that. It's hard.


[User] Please transcribe this text

[Assistant] So how is this done? This is done with convolutional networks. The input to which is a raw image. Here's an input on the left of a number three. And I'll talk about through convolutional layers.


[User] Please transcribe this text

[Assistant] That image is processed, passed through, convolutional layers maintain spatial information.


[User] Please transcribe this text

[Assistant] On the output, they, in this case predicts which of, uh, the images, uh, what number is shown in the image, 0, 1, 2, through 9.


[User] Please transcribe this text

[Assistant] And so this, these networks, that is, this is exactly, everybody is using the same kind of network to determine exactly that. Input is an image, output is a number.


[User] Please transcribe this text

[Assistant] Uh, and in the case of, you know, probability that it's a leopard, what is that number?


[User] Please transcribe this text

[Assistant] Then there's segmentation built on top of these convolutional neural networks where you chop off the end.


[User] Please transcribe this text

[Assistant] Convolutionize the network. You chop off the end where the output is a heat map.


[User] Please transcribe this text

[Assistant] You can do a cat heat map. Where it's.


[User] Please transcribe this text

[Assistant] The part of the image of the output heatmap gets excited, the neurons on that output get excited, and the spatially, spatially excited, uh, in the parts of the image that contain a tabby cap.


[User] Please transcribe this text

[Assistant] And this kind of process can be used to segment the image into different objects. A horse, so the original input on the left is a woman and a horse and the output is a fully segmented image of knowing where's the woman, where's the horse.


[User] Please transcribe this text

[Assistant] And this kind of process can be used for object detection, which is the task of detecting an object in an image.


[User] Please transcribe this text

[Assistant] Now, the traditional method with convolutional neural networks, and in general in computer vision, is the sliding window approach. We have a detector, like the leopard detector, that you slide through the image to find where in that image is the leopard.


[User] Please transcribe this text

[Assistant] This, the segmenting approach, the RCNN approach, is efficiently segment the image in such a way that it can propose different parts of the image that are likely to have a leopard, or in this case a cowboy.


[User] Please transcribe this text

[Assistant] And that drastically reduces the computational requirements of the, of the object detection task.


[User] Please transcribe this text

[Assistant] And so, these networks, this is the, uh, currently one of the best networks for the ImageNet task of localization, is the deep residual networks. They're deep.


[User] Please transcribe this text

[Assistant] So VGG19 is one of the famous ones. VGGnet.


[User] Please transcribe this text

[Assistant] You're starting to get above 20 layers in many cases. 34 layers is the rosinette one.


[User] Please transcribe this text

[Assistant] So the lesson there is the deeper you go,


[User] Please transcribe this text

[Assistant] The more representational power you have, the higher accuracy. But you need more data.


[User] Please transcribe this text

[Assistant] Other applications, colorization of images. So this again,


[User] Please transcribe this text

[Assistant] Input is a single image, and output is a single image.


[User] Please transcribe this text

[Assistant] So you can take a black and white video from a film, from an old film, and recolor it.


[User] Please transcribe this text

[Assistant] And all you need to do to train that network in a supervised way is provide modern films and convert them to grayscale. So now you have arbitrarily sized datasets.


[User] Please transcribe this text

[Assistant] With very little effort on top of it to successfully, well, somewhat successfully recolor images.


[User] Please transcribe this text

[Assistant] Again, Google Translate does image translation in this way. Image to image.


[User] Please transcribe this text

[Assistant] Famous German, correct me if I'm wrong. Uh, dark chocolate written in German on a box. So this can take this image, detect the different letters, convert them to text, translate the text, and then using the image to image mapping,


[User] Please transcribe this text

[Assistant] Map the letters, the translated letters back onto the box, and you can do this in real time, uh, on video.


[User] Please transcribe this text

[Assistant] So, what we've talked about up to this point, on the left are Vanilla Neural Networks. Convolutional Neural Networks. That map a single input to a single output. A single image to a number. Single image to another image.


[User] Please transcribe this text

[Assistant] Then there is recurrent neural networks that map, this is the more general formulation, that map a sequence of images, or a sequence of words, or a sequence of any kind to another sequence. And these networks are able to do incredible things with natural language, with video,


[User] Please transcribe this text

[Assistant] And any time series data. For example, we can convert text to handwritten digits with handwritten text. Here we type in,


[User] Please transcribe this text

[Assistant] And you could do this online, type in deep learning for self driving cars and it will use an arbitrary handwriting style to generate the words deep learning for self driving cars. This is done using recurrent neural networks. We can also take


[User] Please transcribe this text

[Assistant] Car RNNs, they're called, it's character level recurrent neural networks that train on a data set, an arbitrary text data set, and learn to generate text one character at a time.


[User] Please transcribe this text

[Assistant] So there is no preconceived, syntactical, semantic structure that's provided to the network. It learns that structure.


[User] Please transcribe this text

[Assistant] So, for example, you can train it on Wikipedia articles, like in this case.


[User] Please transcribe this text

[Assistant] And it's able to generate successfully not only text that makes


[User] Please transcribe this text

[Assistant] Some kind of, uh, grammatical sense at least.


[User] Please transcribe this text

[Assistant] But also keep perfect syntactic structure for Wikipedia, for Markdown, editing, for LaTeX editing, and so on.


[User] Please transcribe this text

[Assistant] This text says, naturalism and decision for the majority of Arab countries, capitalized, whatever that means, was grounded by the Irish language by John Clare, and so on.


[User] Please transcribe this text

[Assistant] These are sentences, if you didn't know better, that might sound correct. And it does so when you pause, one character at a time.


[User] Please transcribe this text

[Assistant] So, there is, these aren't words being generated. This is one character. You start with the beginning three letters, nat. You generate you completely without knowing of the word naturalism. This is incredible.


[User] Please transcribe this text

[Assistant] Start a sentence and let the neural network complete that sentence. So for example, if you start the sentence with, life is, or life is about, actually,


[User] Please transcribe this text

[Assistant] It will complete it with a lot of fun things. The weather. Life is about kids. Life is about the true love of Mr. Mom. It's about the truth now.


[User] Please transcribe this text

[Assistant] And this is from, uh, Jeffrey Hinton. The last two. If you start with the meaning of life.


[User] Please transcribe this text

[Assistant] It can complete that with the meaning of life as literary recognition. May be true for some of us here.


[User] Please transcribe this text

[Assistant] And the meaning of life is the tradition of ancient human reproduction. Also true for some of us here, I'm sure.


[User] Please transcribe this text

[Assistant] Okay, so what else can you do? You can, uh, this has been very exciting recently is image caption recognition. Ah, generation, I'm sorry.


[User] Please transcribe this text

[Assistant] Image capture generation is important for large data sets of images where we want to be able to determine what's going on inside those images. Especially for search. If you want to find a man sitting on a couch with a dog, you type it into Google and it's able to find that.


[User] Please transcribe this text

[Assistant] So here shown in, in black text, a man sitting on a couch with a dog is generated by the system. A man sitting on a chair with a dog in his lap is generated by a human observer.


[User] Please transcribe this text

[Assistant] And again, these annotations are done by detecting the different obstacles, uh, the different objects in the scene. So segmenting the scene, detecting on the right there's a woman, a crowd, a cat, a camera holding, you know, purple. All of these words are being detected.


[User] Please transcribe this text

[Assistant] Then a syntactically correct sentence is generated, a lot of them, and then you order which sentence is the most likely. And in this way you can generate very accurate, uh, labeling of the images.


[User] Please transcribe this text

[Assistant] Captions for the images. And you can do the same kind of process for image question answering.


[User] Please transcribe this text

[Assistant] You can ask how many, so quantity, how many chairs are there? You can ask about location, where, where are the right bananas?


[User] Please transcribe this text

[Assistant] You can ask about the type of object. What is the object on the chair? It's a pillow.


[User] Please transcribe this text

[Assistant] Uh, you could do the same thing with, uh, with, uh, video caption generation. Video caption description generation. So looking at a sequence of images as opposed to just a single image.


[User] Please transcribe this text

[Assistant] What is the action going on in this, uh, in this situation?


[User] Please transcribe this text

[Assistant] This is the difficult task. There's a lot of work in it, uh, in this area. Uh, on the left is the correct description, so a man is doing stunts on his bike. I heard of zebras that are walking in the field. And on the right, there's a small bus running into a building.


[User] Please transcribe this text

[Assistant] Relevant entities, but just doing an incorrect description. Uh, a man is cutting a piece of, a piece of a pair of a paper.


[User] Please transcribe this text

[Assistant] It's cutting a piece of a pair of a paper. So the words are correct. Perhaps. But, so you're close.


[User] Please transcribe this text

[Assistant] But, uh, no cigar. So one of the interesting things...


[User] Please transcribe this text

[Assistant] If you think about the way we look at images, human beings look at images. We only have a small phobia with which we focus on in a scene. So right now your periphery is very distorted. The only thing, if you're looking at the slides, or you're looking at me, that's the only thing that's in focus. Majority of everything else is out of focus.


[User] Please transcribe this text

[Assistant] So we can use the same kind of concept to try to teach a neural network to steer around the image. Both for perception and generation of those images. This is important first on the general artificial intelligence point of it being just fascinating.


[User] Please transcribe this text

[Assistant] That we can select, selectively steer our attention, but also it's important for things like drones that have to fly at high speeds in an environment where 300 plus frames a second you have to make decisions. So you can't possibly localize yourself or perceive the world around yourself successfully if you have to interpret the entire scene.


[User] Please transcribe this text

[Assistant] So what you can do is you can steer, for example here shown is reading, reading, uh, house numbers, uh, by steering around an image.


[User] Please transcribe this text

[Assistant] You could do the same task for reading and for writing. So reading numbers here on the MNIST dataset on the left is reading numbers, but you can also selectively steer a


[User] Please transcribe this text

[Assistant] Higher resolution as the steering goes on.


[User] Please transcribe this text

[Assistant] That would, drumstick hitting that particular object makes. So you can get texture information from that uh, impact.


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[Assistant] So, here is a video of a human soccer player playing soccer and a state of the art machine playing soccer.


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[Assistant] And, well, let me give him some time to build up.


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[Assistant] Okay. So, uh, soccer, this is, we take this for granted, but walking is hard.


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[Assistant] Object manipulation is hard. Soccer is harder than chess for us to do. Much harder.


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[Assistant] On your phone now, you have, you can have a chess engine that beats the best players in the world.


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[Assistant] Because the question is, and this is a painful video, the question is, where does driving fall? Is it closer to chess or is it closer to soccer?


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[Assistant] For those incredible, brilliant engineers that worked on the most recent DARPA challenge, this would be a very painful video to watch, I apologize.


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[Assistant] With the basic object manipulation and walking tasks.


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[Assistant] So, it's mostly a fully autonomous navigation task.


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[Assistant] Maybe I'll just let this play for a few minutes, for a few moments to...


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[Assistant] To let it internalize how difficult this task is. Of balancing.


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[Assistant] Of planning in an underactuated way where you don't have full control of everything. When, when there is a delta between your perception of what you think the world is and what the reality is. So there, a robot was trying to turn an object that wasn't there.


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[Assistant] And this is an MIT entry that actually successfully, I believe, gotten points for this because it-


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[Assistant] But as, as a lot of the teams talked about, the hardest part, so one of the things the robot had to do is get into a car and drive it and get out of the car.


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[Assistant] So in our evaluation of what is difficult about driving, we have to remember


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[Assistant] That some of those things we may take for granted in the same kind of way that we take walking for granted.


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[Assistant] Let me just quickly read that quote. And quote in the large, highly evolved sensory and motor portions of the human brain is billions of years of experience about the nature of the world and how to survive in it. So this is data. This is big data. Billions of years. And abstract thought, which is reasoning, the stuff we think is intelligence, is perhaps less than 100,000 years, uh, of data. Old.


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[Assistant] We, it hasn't, it's been, it's been very recent that we've learned how to think. And so we respected perhaps more.


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[Assistant] Then the things we take for granted like walking and visual perception and so on. But those may be strictly a matter of data. Data and training time. And network size.


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[Assistant] And that's an important question because the margin of error is small.


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[Assistant] 100 million miles. That's the number of people that die in car crashes every year. One fatality per 100 million miles.


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[Assistant] That's a .0000001% margin of error. That's through all the time you spend on the road, that is the error you get. We're impressed with ImageNet being able to classify a leopard, a cat, or a dog at close to


[User] Please transcribe this text

[Assistant] Uh, at above human level performance. But this is the margin of error we get with driving.


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[Assistant] And we have to be able to deal with snow, with heavy rain, with big open parking lots, with parking garages, any pedestrians that behaves irresponsibly, as rarely as that happens.


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[Assistant] Or just unpredictably, again, especially in Boston, reflections.


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[Assistant] The ones especially, this is one of the things you don't think about, the lighting variations that blind the cameras.


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[Assistant] The question was, if that number changes, if you look at just crashes. So fatalities per crash.


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[Assistant] So the number of crashes is much, much larger than the number of fatalities, which is a great thing. We've built safer cars, uh, but still, you know, even one fatality is too many.


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[Assistant] Is uh, quite open about their performance since hitting public roads. This is from a report that shows the number of times


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[Assistant] The driver disengage, uh, the, the car gives up control.


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[Assistant] That it asks the driver to take control back or the driver takes control back by force.


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[Assistant] Meaning that they're unhappy with the decision that the car was making or it was putting the car or other pedestrians or other cars in unsafe situations. And so if you see over time that there's been a total from 2014 to 2015


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[Assistant] There's been a total of 341 times on beautiful San Francisco roads. Uh, and I say that seriously because the weather conditions are great there.


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[Assistant] Uh, 341 times that the driver had to elect to take control back. So it's a work in progress. Let me give you something to think about here. This, with neural networks, is a big open question. The question of robustness.


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[Assistant] So, this is an amazing paper. I encourage people to read it. There's a couple papers around this topic. Deep neural networks are easily fooled. So, here are eight images.


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[Assistant] Where, if given to a neural network as input, a convolutional neural network as input.


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[Assistant] The network with higher than 99.6% confidence says that the image, for example, on the top left is a robin.


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[Assistant] Next to it is a cheetah, then an armadillo, a panda, an electric guitar, a baseball, a starfish, a king penguin.


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[Assistant] All of these things are obviously not in the images. So networks can be fooled with noise.


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[Assistant] More importantly, more practically for the real world.


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[Assistant] Adding just a little bit of distortion, a little bit of noise distortion to the image.


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[Assistant] So here's an example, uh, there's three columns. Correct image classification, the slight addition of distortion,


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[Assistant] And the resulting prediction of an ostrich for all three images on the left.


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[Assistant] And a prediction of an ostrich for all three images on the right.


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[Assistant] This ability to fool networks easily brings up an important point. And that point is that, that there has been a lot of excitement


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[Assistant] About neural networks throughout their history. There's been a lot of excitement about artificial intelligence throughout its history. And...


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[Assistant] Uh, not coupling that excitement, not grounding that excitement in the reality, the real challenges around that has resulted in, in crashes. In AI winters, when funding dried out.


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[Assistant] And people became hopeless in terms of the possibilities of artificial intelligence. So here's a 1958 New York Times article.


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[Assistant] That said the Navy revealed the embryo of an electronic computer today. This is when the first perceptron that I talked about was implemented in hardware by Frank Rosenblatt. It took 400 pixel image input and it provided a single output.


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[Assistant] Weights were encoded in hardware potentiometers and weights were updated with electric motors.


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[Assistant] Now New York Times wrote, the Navy revealed the embryo of an electronic computer today.


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[Assistant] That expects we'll be able to walk, talk, see, write, reproduce itself, and be conscious of its existence.


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[Assistant] Dr. Frank Rosenblatt, a research psychologist at the Cornell Aeronautical Laboratory, Buffalo, said perceptrons might be fired to the planets as mechanical space explorers.


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[Assistant] This might seem ridiculous, but this was the general opinion of the time.


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[Assistant] And as we know now, perceptrons cannot even separate a nonlinear function.


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[Assistant] They're just linear classifiers. And so this led to two major AI winters in the 70s and the late 80s and early 90s.


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[Assistant] The Lighthill Report in 1973.


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[Assistant] Beyond the capabilities of our research, reports like this will come and they have


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[Assistant] The possibility of creating another AI winter. So I want to pair the optimism, some of the cool things we'll talk about in this class, with the reality of the challenges ahead of us.


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[Assistant] The focus of the research community. This is some of the key players in deep learning.


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[Assistant] What are the things that are next for deep learning? The five-year vision.


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[Assistant] We want to run on smaller, cheaper mobile devices.


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[Assistant] We want to explore more in the space of unsupervised learning, as I mentioned, and reinforcement learning.


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[Assistant] Explore the space of videos more, the recurrent neural networks, like being able to summarize videos, or generate short videos.


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[Assistant] One of the big efforts, especially in the companies dealing with large data, is multimodal learning. Learning from multiple data sets with multiple sources of data. And lastly, making money from these technologies. There's a lot of, this, despite the excitement,


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[Assistant] There has been an inability for the most part to make serious money from some of the more interesting parts of deep learning.


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[Assistant] And while I, uh, got made, made fun of by the TAs for including this slide because it's shown in so many sort of business type lectures, but it is true that we're at the peak of a hype cycle and we have to make sure


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[Assistant] Given the large amount of hype and excitement there is, we proceed with caution.


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[Assistant] Is, we already talked about spoofing the cameras, spoofing the cameras with a little bit of noise. So if you think about it,


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[Assistant] Self-driving vehicles operate with a set of sensors.


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[Assistant] And they rely on those sensors to convey, to accurately capture that information.


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[Assistant] Now what happens, not only the world itself produces noisy visual information, but what if somebody actively tries to spoof that data?


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[Assistant] One of the fascinating things that have been recently done is spoofing of lidar.


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[Assistant] So these, LiDAR is a range sensor that gives a 3D point cloud of the objects in the external environment. And you're able to successfully


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[Assistant] Do a replay attack, where you have the car.


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[Assistant] See people and other cars around it when there's actually nothing around it.


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[Assistant] In the same way that you can spoof a camera to see things that are not there. A neural network.


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[Assistant] So let me run through some of the libraries that we'll work with and that are out there that you might work with if you proceed with deep learning. TensorFlow, that is the most popular one.


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[Assistant] These days, it's heavily backed and developed by Google.


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[Assistant] It has a, eh, it's primarily a Python interface, uh, and is,


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[Assistant] Uh, very good at, at operating on multiple GPUs.


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[Assistant] There's Keras and also TF Learn and TF Slim, which are libraries that operate on top of TensorFlow that make it slightly easier, slightly more user friendly interfaces to get, get up and running.


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[Assistant] If you're interested to get in, uh, at the lower level, uh, tweaking of the different parameters of neural networks, uh, creating your own architectures, Torch is excellent for that, with its own, uh, Lua interface, Lua is a programming language, and heavily backed by Facebook.


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[Assistant] It's one of the first libraries that supported, that came with GPU support. It definitely encourages lower level tinkering. And it's a Python interface.


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[Assistant] For doing some of the low level computations involved with training these neural networks on NVIDIA GPUs. MXNet, heavily supported by Amazon.


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[Assistant] And uh, they've officially recently announced that they're going to be, their AWS going to be all in on uh, MXNet.


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[Assistant] Cafe, started in Berkeley, uh, also was very popular in Google before TensorFlow came out.


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[Assistant] It's primarily designed for computer vision with conv nets, but it's now expanded to, uh, all the, uh, all other domains.


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[Assistant] There is CNTK as it used to be known and now called the Microsoft Cognitive Toolkit. Nobody calls it that still, I'm aware of.


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[Assistant] It says multi GPU support, has its own brain script, custom language, as well as other interfaces.


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[Assistant] And what we'll get to play around in this class is amazingly deep learning in the browser, right?


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[Assistant] Our favorite is ComNet.js, what you'll use, built by Andre Karpathy, from Stanford, now OpenAI.


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[Assistant] It's good for explaining the basic concept of neural networks. It's fun to play around with. All you need is a browser, so very few requirements. It can't leverage GPUs, unfortunately.


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[Assistant] But for a lot of things that we're doing, you don't need GPUs. You'll be able to train a network with very little and relatively efficiently without the need of GPUs. It has full support for CNNs, RNNs, and even deeper reinforcement learning. Keras.js, which seems incredible, we tried to use for this class.


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[Assistant] It didn't happen to, uh, it has GPU support. So it runs in the browser with GPU support with OpenGL or however it works, uh, magically. But we're able to accomplish a lot of things we need without the use of GPUs.


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[Assistant] I should, it's incredible to live in a day and age when uhm,


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[Assistant] It literally, as I'll show in the tutorials, it takes just a few minutes to get started with building your own neural network that classifies images.


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[Assistant] And a lot of these libraries are friendly in that way. So all the references mentioned in this


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[Assistant] Presentation are available at this link and the slides are available there as well.


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[Assistant] So I think in the interest of time, let me wrap up. Thank you so much for coming in today. And tomorrow I'll explain the deep reinforcement learning game and the actual competition and how you can win it. Thanks very much guys.


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[Assistant] Okay, so what problem are we trying to solve?


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[Assistant] Various forces and equation that we need.


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[Assistant] So that's the problem we looked at. This is F sub 0. I may have just called it F last time. Doesn't matter. It's a matter of these symbols. Okay?


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[Assistant] I'm using the fact that omega naught was defined as root of k over n. We used to call it omega.


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[Assistant] But now I'm calling it omega naught, and you should understand why I'm doing that. Because there are now two omegas in the problem.


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[Assistant] That is the omega of the driving force. So you should imagine an oscillatory force grabbing this mass and moving it back and forth.


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[Assistant] Omega naught is the natural frequency of the mass. If you pull the spring and let it go, that's the frequency that it will vibrate. There are really natural frequencies which is fixed by k and m, and omega is for you to decide. You can decide to apply to it any force you like of any frequency.


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[Assistant] So first take the easy case in which the right hand side is zero. No driving force.


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[Assistant] That's the problem I spent considerable amount of time on yesterday. You can say, well, if there's no driving force, why does anything move?


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[Assistant] That's actually correct. If you took a mass and a spring and you didn't do anything to it, the solution to this equation is x equal to zero, and that's just fine.


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[Assistant] But it also has interesting non-trivial solutions because you can pull the mass and then let it go. After you let it go,


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[Assistant] That are no forces on the system, except the force of friction and the force of the spring. And that is this problem.


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[Assistant] How do you solve this problem? I told you the way you solve this problem is you make a guess. X of t is some number a times e to the alpha t. See if it'll work.


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[Assistant] And you find it'll work and you can do this in your head. I certainly don't want to repeat all of that.


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[Assistant] The reason it's going to work is that every time you take a derivative of x, you bring down an alpha.


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[Assistant] And still keep e to the alpha t. So when you put that guess into this equation, you're going to get alpha squared plus gamma alpha plus omega naught squared times A e to the alpha t equal to 0.


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[Assistant] And if this is going to vanish, it's not because of a, and it's not because of e to the alpha t, it's because this guy in brackets is going to vanish. So that's zero.


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[Assistant] You all recognize this to be a quadratic equation. You go back to your


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[Assistant] These are the roots. Now here you have to look at


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[Assistant] So anyway, what does this mean? This means that yes, you can find a solution to this equation that looks like Ae to the alpha t. A can be any number you like.


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[Assistant] No restriction was placed on a, but alpha cannot be any number you like. Alpha has to be the solution of this quadratic equation, and there are two solutions. So you can call them alpha plus and alpha minus.


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[Assistant] That's when I branched off into two different cases. Case one, you can see, look, when you do quadratic equations, you know the


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[Assistant] The weight going to fall out is depending on whether the stuff inside the bracket is positive or negative.


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[Assistant] So let's take the easier case when it's positive. That means gamma over 2 is bigger than omega naught. Gamma, you remember, is the force of friction, so we want it to be


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[Assistant] Not only non-zero but bigger than this amount controlled by the natural frequency. In that case, let's not go into details, but you can all see


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[Assistant] Uh, you put whatever value you have for γ, the square root of this is some number, the whole thing is going to be two real choices you get.


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[Assistant] 1 will be minus gamma over 2 plus


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[Assistant] Whatever is inside the square root, other will be minus gamma over 2 minus what you have in the square root.


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[Assistant] The only thing I want you to notice is that both these roots are negative.


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[Assistant] This guy is obviously negative because it's a minus gamma over 2 minus something else. That's definitely negative. You might be a little worried here that this is minus gamma over 2 and this is plus something.


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[Assistant] But the plus something is smaller than the minus something because what's inside the square root is gamma over 2.


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[Assistant] Square, if you took the square root of that, you will get exactly gamma over 2, but you are taking away from it this number. Therefore, this square root is bigger, is smaller than this number, so the overall thing will still come out negative.


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[Assistant] So why am I so eager that it should come out negative? Why am I eager that the root should be negative? Yes?


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[Assistant] Uh, no, even if I had positive, if I took two derivatives, it will work out. Yes?


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[Assistant] This is not a cosine function. This is never going to be.


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[Assistant] Cosine functions related to e to the i omega t or e to the i x. These are all real exponentials, yes.


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[Assistant] Yeah, that's our main problem. When you write an exponential function,


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[Assistant] Time is the absolute value of alpha, just to show you it's a falling exponential. And we have a stomach only for falling exponentials. You gotta think. If I pull this mass and I let it go,


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[Assistant] I expect x to go down from whatever value I had, not to keep growing indefinitely. That doesn't make any physical sense.


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[Assistant] And one of the great things about the union of mathematics and physics is the equation


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[Assistant] Automatically works out that way. It gives you an answer that makes sense.


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[Assistant] Okay, so what you get then is a solution that looks like this. One number a times e to the one of the roots, uh, plus another number b times e to the other root.


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[Assistant] I wrote them, they're both e to the alpha t, but I write it as minus the absolute value, so you can all see their falling exponentials. You understand if a number is negative, like negative 5,


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[Assistant] That can be written as the minus 1 times the absolute value of the number.


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[Assistant] So it is e to the alpha t, but I want to write it in this form. So it's falling.


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[Assistant] So this is a case where if you pull the mask and you let it go, it will just relax.


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[Assistant] Eventually coming to x equal to zero, which is its normal location, equilibrium location.


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[Assistant] Now how about the numbers A and B? Right now they are completely arbitrary.


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[Assistant] In order to find A and B, we need to be given more information other than simply the mass and the spring constant.


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[Assistant] They're all sitting here and the friction coefficient, but that's not enough to tell you what it does on a given day. It wants to know, well, yes, it's the mass, it's that spring, but did you pull it or did you release it in this way or that way? That's what will control the answer.


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[Assistant] They are called initial conditions. One initial condition is, what was the value of x at time t equal to 0? That's asking you how much you pulled it when you let it go.


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[Assistant] The other initial condition is, what is the velocity, namely the derivative of x at time zero.


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[Assistant] Most of the time what you will find is you pull the mass and you release it, then it's got a non-zero x and a zero velocity.


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[Assistant] But it doesn't have to be that way because I could have pulled it and released it at t equal to zero, starting to move inwards, and that's when you jump in and you set your clocks to zero.


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[Assistant] Well, at that instant, x has a value, and the velocity also has a non-zero value. They don't all have to be zero, but they are some two numbers you can pick at will. You can see that if I put t equal to zero, I'm going to get


[User] Please transcribe this text

[Assistant] Times z to the 0, which is 1. I'm going to get a plus b is x0.


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[Assistant] That's one condition on A and B. Now take a derivative, namely find the velocity, then put t equal to 0.


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[Assistant] And set that equal to some pre-assigned number, you'll get a second equation for A and B, and we all know we can solve those two equations.


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[Assistant] To find a and b. That's how we fit these three parameters


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[Assistant] To the solution Taylor made for that particular situation.


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[Assistant] I can pick these numbers, x0 and v0, but why not keep going? Maybe the acceleration at time 0 is some number. Why is that not up to me to decide?


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[Assistant] Can anyone tell me why that's not negotiable? Yes?


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[Assistant] Can I pick the acceleration of a body at will? Yes?


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[Assistant] Yeah, I'm not sure I heard the exact answer, but it sounds close enough to what I had in mind, which is Newton will tell you what the acceleration of a body is. Namely, mass times acceleration is the force, and the force, once it's given in any given problem, which is given by where the body is,


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[Assistant] And what the velocity is in this example, you have no choice on what the acceleration is.


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[Assistant] Newton's law says if the force is so-and-so, the acceleration is so-and-so, but he doesn't tell you what should be the velocity at a given time, what should be the position at a given time. That's why there are two free numbers.


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[Assistant] That's why you have the answer with two free numbers that you can pick to bring it in accord with these two numbers.


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[Assistant] Okay, this solution to the equation with x equal to 0, from now on, I'm going to call x sub c, where c stands for the complementary function.


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[Assistant] Now there's another thing that's going to happen here, which is in fact


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[Assistant] Another alternative, let me, you can think about what I'm saying.


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[Assistant] The other alternative is when this gamma over 2 is smaller than this omega naught. Then, of course, you have the square root of a negative number.


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[Assistant] To the state where you're finding the root, and you say alpha equals minus gamma over 2.


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[Assistant] Plus or minus square root of gamma over 2 squared minus omega naught squared, but that number is negative.


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[Assistant] So that can be written as a negative sign times the square root of this number.


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[Assistant] And the square root of minus 1 will come out as an i. So either you will get two real roots.


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[Assistant] This is a mathematical result, nothing to do with physics. A quadratic equation with real coefficients either will have two real roots,


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[Assistant] Or it will have two roots which are complex conjugates of each other.


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[Assistant] You have learned the notion of complex conjugation, which is the change i to minus i. You can see the α± and α± have the same real part, minus γ over 2, and have opposite imaginary parts. So in this example, α± is the complex conjugate of α±.


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[Assistant] And I've told you this relationship works both ways. That for alpha minus is a complex conjugate of alpha plus.


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[Assistant] Omega prime t, but I'm going to call this fellow in brackets as omega prime.


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[Assistant] Then there's the second solution. The real part is exactly the same. The imaginary part is opposite, minus iω't.


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[Assistant] Omega prime is a new frequency connected to the natural frequency and the frictional force. That's why we've got three omegas. We've got to keep that in mind. Omega naught is a natural frequency vibration.


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[Assistant] If I have a driving force, I reserve the symbol omega for the frequency of the driving force.


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[Assistant] Here is yet another quantity with units of frequency, which I'm calling ω'


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[Assistant] So what does this guy look like? Well first of all it's full of complex numbers and we don't like it.


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[Assistant] We know the oscillator should have a real coordinate, so you ask, I've not picked a and b yet, how shall I pick them so x is real?


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[Assistant] Well, forget A and B for a minute and look at this number and this number. Do you guys realize that they're complex conjugates of each other? Because i has gone to minus i. So, for example, if A and B were equal numbers,


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[Assistant] This would be a real number, because number plus its conjugate, you kill the imaginary part, because they have opposite imaginary parts. So A and B, real, is a good answer, but it's too restrictive. All we will demand


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[Assistant] Instead of a, it looks like absolute value of a times e to the i phi.


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[Assistant] I remind you, in the complex plane, if a number z is sitting there, the complex conjugate is sitting here, they have opposite angles.


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[Assistant] Absolute value, e to the minus gamma over 2 times e to the i omega prime t plus


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[Assistant] A and B both being fully real means this phi is zero.


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[Assistant] But you see, phi doesn't really have to be 0. That is too much of a restriction.


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[Assistant] Phi can be whatever you like, provided the phi that A has, the phase that A has, same as the phase that B has. That means the angle in the complex plane of the number A, the number B, have to be opposite of each other.


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[Assistant] You guys should know what it is. I've told you many times this is unforgivable if you don't know what this is. e to the iθ plus e to the minus iθ over 2 is cosθ. Since you don't have the over 2, you can write it as 2a


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[Assistant] E to the minus gamma over 2, cosine of whatever is in here, omega prime t plus phi.


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[Assistant] In view of what's to come, instead of calling this phi, I'm going to call it phi 0.


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[Assistant] So 2 times a is a new number. I'm going to call that c. And this is the bottom line. Answer looks like e to the minus gamma t over 2, cosine omega prime t plus phi 0.


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[Assistant] So we've done all this yesterday. I've given you the notes, but I want you to understand


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[Assistant] Where we are going with all of this? If you pull a mass and let it go, it turns out


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[Assistant] There are two things that can happen. I really wish I had time to bring you these different masses and show you.


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[Assistant] If you have no friction, everyone knows what's going to happen. You pull the mass and let it go, it oscillates forever.


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[Assistant] In real life, there is always some friction. So let's turn on a little friction. What will happen is what's very familiar and expected. We'll start out with an amplitude of 5.


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[Assistant] Go to four, go to three, go to two, rock back and forth, and slowly, it may wiggle around, but you won't be able to see it. That's called a damped oscillation. That's really what's described by this solution.


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[Assistant] Here, it looks like a cosine of something, but the number in front of it itself is falling with time.


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[Assistant] And the picture for that will look like the following.


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[Assistant] An exponentially falling function as your amplitude, and you multiply that by an oscillating cosine, it will do this. This is a damped oscillation.


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[Assistant] But another extreme case which you may not think about is, if you crank up the friction more, till gamma over 2 is bigger than omega naught,


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[Assistant] Then what happens is, if you pull the mass and let it go, it doesn't even go to the other side of x equal to 0.


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[Assistant] It just slowly gets to zero in an infinite amount of time. You understand? If you pull it from here, you let it go, no friction is this for eternity.


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[Assistant] Small amount of friction will slowly damp the oscillation.


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[Assistant] Too much of friction, you come back slowly and you come to rest.


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[Assistant] These are the different cases, and the mathematics tells you the different cases. So that's what we like about the union of mathematics and physics. Once you wrote down the basic law of Newton, you have no choice as to where it takes you. You go where it takes you. These are the different things.


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[Assistant] Okay, so this is what I did earlier on in the day, but now let's go back.


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[Assistant] So let's go back to the more general problem I had, which is a right-hand side which is not zero. So let's write the right-hand side to not zero, x double dot plus gamma x dot plus omega naught squared x equals f naught over m cos omega t.


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[Assistant] This is called the driven oscillator. The driven oscillator is present all the time. For example, if you have a swing, swing goes back and forth.


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[Assistant] Whereas in the absence of external forces, all oscillations will eventually die down. You guys understand that?


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[Assistant] Why? Why do all oscillations die down in the absence of external forces?


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[Assistant] Yeah, well in this case it's not air resistance but friction. But what friction does is that in the presence of friction you cannot prove a law of conservation of energy. We saw that. Whenever friction acts, the energy, total mechanical energy goes down.


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[Assistant] So kinetic plus potential will decrease with time, so the oscillations get damp.


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[Assistant] But if you are pushing it from outside, you can provide a constant source of energy so you can maintain a steady oscillation. And that's the problem we're looking at. That is this problem.


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[Assistant] This problem, well I said I don't want to solve this equation, I'm going to manufacture another problem. z double dot plus gamma times z dot.


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[Assistant] So, let's be very clear. This is the real physics problem that occurs all the time.


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[Assistant] This is clearly not of our world, because in our world


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[Assistant] You don't have a force e to the i omega t. All forces in the real world are real.


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[Assistant] So this is a complex force. If you break down the e to the i omega t,


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[Assistant] You all know it's cos omega t, which I like, plus i sine omega t, which I don't like.


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[Assistant] I'm solving a problem in which on top of the force I'm really applying, that's the imaginary force put on top of it.


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[Assistant] So, what is the strategy? Why are we taking the problem and making it worse than it is? Because we have a strategy. The strategy is the following.


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[Assistant] If you have right hand side cos omega t, ask yourself what you're trying to do. You're looking for some function x of t, which will obey this equation. You can try various functions, maybe x is a cosine.


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[Assistant] I'm sorry, no derivative will be a cosine, one derivative will be a sine. So you cannot make it into a cosine.


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[Assistant] If you took a sine, the middle guy will look like a cosine. You can try to balance that against this. The end two terms look like cosines. So neither a sine nor a cosine works. Now it turns out something in between will work.


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[Assistant] But something that always works is the following. If the right hand side, instead of being cosωt, was e to the iωt, you can guess the answer immediately.


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[Assistant] Namely, z itself will behave like e to the iωt, because then if you take two derivatives, that's going to look like e to the iωt times some numbers, that's going to look like e to the iωt, that's going to look like e to the iωt. So the beauty of the exponential, that when you take derivatives, you keep getting the exponential.


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[Assistant] Okay. But now, the problem is, this is not the problem. No one told you to solve this problem. But our strategy is the following.


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[Assistant] The right hand side has got a real force and imaginary force. The real force will produce the x that I want. The imaginary force, purely imaginary, i times an omega t, will produce a purely imaginary answer. So at the end of the day, I will find the answer and keep only the real part. That's the plan. That's going to be the strategy. So that this z,


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[Assistant] This d is going to be the x that I want, and we'll have a part hanging around y that I don't want, but y is being driven by the sine omega t, and x is being driven by the cos omega t. But now let's solve it by inspection.


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[Assistant] So, let's make a guess that z looks like some number z0 e to the i omega t.


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[Assistant] Now it's very easy to take derivatives of exponential. Just pull down i-omega every time you see a derivative.


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[Assistant] So z double dot will become i omega squared, which is minus omega squared.


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[Assistant] Z dot will become I omega, there's a gamma there, and omega naught squared is just multiplying by a number. So this is what I get on the left hand side, and I want it to equal this on the right hand side.


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[Assistant] And you're almost there, because the time dependence of the two sides are identical, so you can cancel it.


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[Assistant] And what you find is yes, I can satisfy the equation that's given to me, provide the number z0, the number in front of it, obeys this condition.


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[Assistant] That's trivially solved, because we all know how to get the solution to that one. z0 is obtained by dividing both sides by everything that multiplies z0, which happens to be ω0 squared minus ω squared plus iω gamma.


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[Assistant] So the beautiful property of the exponential function is that you take what is a differential equation and turn it into an algebraic equation.


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[Assistant] Because the act of taking derivatives generally is a nasty operation. Sines become cosines, x cubed becomes 3x squared, but exponential remains exponential. That's the property you're trying to exploit here. So now we've got z naught to be this one.


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[Assistant] And z itself, the function, the complex function looking for, is f0 over m.


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[Assistant] E to the i omega t divided by this quantity downstairs.


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[Assistant] This quantity downstairs I'm going to call I of ω. I of ω is a complex number.


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[Assistant] It's got a real part and an imaginary part. The real part is omega naught squared minus omega squared. Imaginary part is omega gamma.


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[Assistant] It's not a fixed complex number. At every frequency of the driving force Ω, it's a different complex number. But if you're sitting at one frequency of the driving force, it's a particular number. Everyone should know how to work it out.


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[Assistant] Omega 0 is root of k over m. Omega is given to you. That's some real number. Gamma is given to you. That's an imaginary part. If you actually put in numbers, it might look like 6 plus 5i.


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[Assistant] The thing I call impedance, or I, is just a complex number. But it varies from frequency to frequency. You can see that.


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[Assistant] Okay, now, we got ZFT. If the problem given to us was Z, we are done. We don't have to do any more work. Here is the answer.


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[Assistant] Anyone know why that's not the way to get the real part yet? Yep. Yes?


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[Assistant] Yes, this number downstairs is not a real number. It's a complex number.


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[Assistant] So if you have a complex number on the top divided by a complex number on the bottom, the real part is not simply the real part of the top. You've got to write the whole thing so you can clearly separate the real and imaginary parts. And that's what I'm going to do next.


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[Assistant] Every complex number we know can be written as an absolute value times e to the iphone.


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[Assistant] Every z looks like mod z e to the iθ, because then it's mod z cosθ, which is x, plus i times mod z sinθ, which is iy. Every complex number can be written that way. And what will be the magnitude of this guy? Well, by the Pythagoras Theorem, it'll be the square of the real part, plus the square of the imaginary part under root.


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[Assistant] That you have will have an absolute value which is that squared with that square under root.


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[Assistant] And it'll have a phase Φ obeying the condition tan Φ is the imaginary part divided by the real part.


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[Assistant] I want you to realize that in a given problem, with a given mass, given friction, given frequency, both these are known.


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[Assistant] There's simply a magnitude of I, and the phase Φ can be found by this process. Now, write it like that. Then you get F 0 over m divided by absolute value of


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[Assistant] I'm almost there, because e to the iΦ in the bottom can be written as e to the minus iΦ on the top.


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[Assistant] Now it's easy to take the real part because this is cos omega t minus phi plus i sine omega t minus phi. These are all real numbers.


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[Assistant] So they'll just multiply the whole thing, but the real to imaginary separation now comes from just taking the cosine part of this exponential.


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[Assistant] And that is our final answer. That's the answer we've been looking for, which is x of t is equal to f0 over m divided by absolute value of i, cosine omega t minus phi.


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[Assistant] We went there because the answer is not a cosine and it's not a sine.


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[Assistant] So it's got some cosωt, it's got some sinωt, and that's why it's hard to guess their answer. But if you go through complex numbers, it comes out very easy.


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[Assistant] So the next thing is to analyze the properties of the answer you got. You can write this whole thing.


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[Assistant] Notice that even though it's oscillating like a cosine, it's not in step with the applied force.


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[Assistant] The applied force is cos omega t. Its maximum occurs at 0, 2 pi, 4 pi, etc. This guy's maximum occurs somewhat later. The first maximum occurs when omega t is equal to 5. So it's said to be lagging the function. You can imagine two functions oscillating. One is behind the other one. It reaches the maximum slightly later. So the first property is, in a driven oscillator,


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[Assistant] The x will not be in step with the f, it will be lagging, or it could be leading, depends on what phi is.


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[Assistant] Then what about the amplitude of oscillation? That's the thing that's very interesting. How big is the oscillation?


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[Assistant] It looks like f naught over m divided by the absolute value of i. Let me write the absolute value of i for you.


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[Assistant] So the amplitude of oscillation depends on the frequency.


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[Assistant] You've got to draw the graph. As a function of frequency, what does the 1 over this look like? Pardon me?


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[Assistant] Yes, but A minus B squared and B minus A squared. That's why I've not been paying attention.


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[Assistant] Okay, yeah, so I'm pretty careless on what I put in, but this is one occasion where I sort of know it's okay, because I'm going to square this. I don't have to remember who is ω0 and who is ω.


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[Assistant] That goes away, that goes away, you get ω0 squared, and ω0 squared is k over m, and I claim what you will find initially is this will become


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[Assistant] F0 over k. That'll be the value of this whole amplitude at zero frequency.


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[Assistant] I hope that makes a lot of sense to you. Zero frequency means the force is not even oscillating. It's a constant force. What x are you going to get? You're going to get the force divided by the force constant. That's the x to which you will pull the spring.


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[Assistant] If you put all the omegas and masses back into proper place, you'll find it's f0 over k.


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[Assistant] Then what happens is, as omega increases, this denominator, one of these guys in the denominator is starting to vanish.


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[Assistant] If gamma were a very, very tiny number, when omega not equals omega, you would be dividing by a very small number, and the answer will have a huge rise.


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[Assistant] Then you can see when ω goes to infinity, very large ω, you're dividing by ω square, it'll fall off like this.


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[Assistant] Now, one of the homework problems is to find out exactly where it peaks. If gamma is very small, you don't have to do much work to know it peaks at ω, equal to ω0, but otherwise it's near that. That's the phenomenon that's called resonance. Resonance is the following. You are applying a force, F0 cos ωt.


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[Assistant] You go on varying the frequency and response to your force, the system oscillates. The amplitude of oscillation is the largest when the applied frequency is very near the natural frequency.


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[Assistant] If you take the limit of a very small gamma, this maximum is very close to omega naught. That's called resonance.


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[Assistant] If gamma is really tiny, you can get graphs where it's incredibly peak.


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[Assistant] So that's when systems start vibrating a lot. That's why when your car, uh, goes over certain bumps, if they're periodic,


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[Assistant] They are hitting your car at a certain frequency, and if that frequency is anywhere near the natural frequency of vibration of your suspension system in the car, the car begins to resonate.


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[Assistant] That's another example of resonance. People always say, yeah, I can resonate with what the person is saying.


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[Assistant] What does that mean? The person is saying whatever the hell he wants. Some people think it's interesting, some people think it's boring. It's the same thing here. The force is acting at a certain frequency.


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[Assistant] If your natural frequency is close to that, your response is high. If it's low, you don't care. Same thing. That's mechanical systems selecting a frequency. In fact, this is how radios work. You have a radio. There's only one radio station.


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[Assistant] You build an electrical circuits in which the currents like to go back and forth or oscillate at a certain frequency, the radio station should send signals at that frequency, your response will be very high.


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[Assistant] What they do is, each one sends a signal at a different frequency.


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[Assistant] You take your radio and there are dials inside. Well, I forgot. Do you know what a dial is?


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[Assistant] Okay, you are nowadays pushing the, some other button. There was a time when you had to turn a dial. When you turn a dial, you are changing the properties of the electrical circuit and changing its natural frequency.


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[Assistant] You move the natural frequency till it lines up with the frequency of the radio station, with one particular station. Then the response from that station is very high, and that's the sound you hear. Other stations are also present, but you are not near their maximum. Uh, they would, they are, they are sitting somewhere here and somewhere here.


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[Assistant] So you want the station frequency to come under this resonance peak.


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[Assistant] So if you've got a bad radio, it'll pick up sound from many stations because it's not very refined.


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[Assistant] If the peak is broad, you will get sound from many stations. You want to build a system which is very finely sensitive to the frequency.


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[Assistant] That requires gamma to be very small. If gamma is very, very small, you can see if gamma is going to 0, the denominator is


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[Assistant] 1 divided by 0, it's going to be infinite. So if gamma is very tiny, 1 over very tiny is very big, and you can get enormous spikes in response, and that's what people need.


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[Assistant] Okay, the last thing you need to do, this problem set. By the way, I know this problem set is very nasty. Towards the end, it got tough, you got email, uh,


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[Assistant] I agree with you that this, this whole course is somewhat more difficult than it has always been. Okay? I'm aware of that fact. And I will bear this fact in mind when you go into your final.


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[Assistant] So I will remember that in the end at the, when I make the final exam or look at your grade.


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[Assistant] This use of complex numbers is something from which I will not, however, back down.


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[Assistant] It turns out that yes, you can do all of this without using complex numbers. Why? Because in the real world, everything has got real numbers.


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[Assistant] So you can, in fact, by orgy of trigonometric identities, solve it with sines and cosines. But eventually I'll give you a problem where you just cannot do it in a meaningful way.


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[Assistant] The power of complex numbers is they turn differential equations into algebraic equations. All you have to do is find the roots of a quadratic equation.


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[Assistant] And when you do quantum mechanics in second term, complex numbers are in the equations of motion.


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[Assistant] The analog of F equals MA, there's an I sitting there. So you cannot escape the I.


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[Assistant] It's there. So I'm saying, uh, depending on how much exposure you had, you have to do more or less work, but you just got to get used to complex numbers. You got to get used to the absolute value, the phase angle, how to manipulate them.


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[Assistant] If you already know it, good. If you don't know it, you have to do more work than other people. I'm not going to let off on it. Yes.


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[Assistant] Anything you need to know with differential equations is what I've done for you in the class today. That's it. That's the only part you know.


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[Assistant] So you have to go over these steps and work them out. That's the part I cannot do for you. That's something you have to do.


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[Assistant] Alright, what is finally missing is that x of t, if I wrote for you what I wrote above,


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[Assistant] Which is x0, I don't want to write the x0 again, it's f0 over mi, maybe I'll write the whole thing, f0 over m divided by absolute value i times cosine omega t.


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[Assistant] Minus y. It turns out this is not the full answer to the question, what happens when I push a mass? Let's understand why.


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[Assistant] Suppose I go to this mass and spring system, I grab it at t equal to zero and start shaking it. Well, at t equal to zero,


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[Assistant] x was zero, because I grabbed it at t equal to zero. Well at t equal to zero, if I go to this solution, it's not giving me zero. It's saying it's f naught over m i cosine phi.


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[Assistant] So what happened to the freedom I had to assign the velocity and position arbitrarily? Well, it turns out that I can still do that.


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[Assistant] That's because this solution has got a part that's actually missing.


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[Assistant] And the part that's missing is this complementary solution, Ae to the minus gamma t over 2, cosine of omega prime t minus phi naught.


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[Assistant] Because if x obeyed a certain equation, if I put that in, I'm saying adding this part will not change the fact that it obeys the equation.


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[Assistant] Let's ask ourselves why. Go back to the top line there.


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[Assistant] F x double dot plus gamma x dot plus omega naught squared x will be equal to F naught over m cos omega t, if you put this in.


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[Assistant] Now what if you keep adding this to the solution? Will you mess it up? Imagine adding this to the left hand side, but I claim on the left hand side there'll be no extra terms because this guy double dot


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[Assistant] Plus gamma times this guy dot, plus omega naught squared times this guy is zero.


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[Assistant] This is the function which you say is annihilated by the operations on the left hand side. It's like saying the following. If you want a function whose derivative


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[Assistant] And yet, the combination of derivatives appearing in the equation act on this to give you zero.


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[Assistant] That's why you can add it without screwing up what you've done so far. So the way you do these problems is, you first find this function that depends on the applied force, and you add to this solution you get for free, even in the absence of an applied force. Yes?


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[Assistant] This is called a complementary function, and this is called a particular solution. This particular solution depends on the force, and this is the function. I don't know why it's called complementary. Maybe it complements the answer, or it's a freebie and you don't pay for it with any force, but it's called complementary. But you've got to add the two to get the final answer.


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[Assistant] So you will still have these numbers a and Φ0 hanging around, and how do you select them? You select them if on top of it somebody tells you that this driven oscillator had a certain coordinate and certain velocity at t equal to 0. So you'll evaluate the whole expression at t equal to 0.


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[Assistant] By putting t equals 0 everywhere, you'll get one equation involving a and phi-naught.


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[Assistant] And you'll take the derivative for the velocity, set that equal to some given number. You've got a second equation involving a and .0. You'll juggle the two equations and solve for the two numbers, a and .0.


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[Assistant] That's what I tried to do yesterday, but this is not the first time it's happened to me that I tried to go fast. In the end I lost time because I was not happy with the way it was explained. And, uh, I usually come back and say sorry and then do it all over again. Okay? But there's only so much I can do. Uh, the rest is up to you guys. You have to, you have to do the work and you have to do more problems. Okay, so now we are moving off to another topic.


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[Assistant] These are waves. But I think it's going to be, look, I think this probably is the...


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[Assistant] The hardest point of what we're doing in this term, at least mathematically. Everything else we do, heat, hot water, cold water, that kind of stuff you've done in high school, that's all in your future. It's going to be a lot easier. Okay, waves.


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[Assistant] So everyone has a good feeling for waves. You go to a lake, you, you know, if you're a hitman for the mafia, you're dropping things in with some weight. They go down. What's the first thing they do? Once a body is falling into the water,


[User] Please transcribe this text

[Assistant] It's carving out some body of water and diving in, so the water next to it dives in to fill that hole.


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[Assistant] But then the water even next to it drives in to fill the hole left behind by these guys.


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[Assistant] For you, it looks like a bunch of ripples traveling outwards from the center.


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[Assistant] So, this is the classic example of a wave. If you've got one example, you can latch onto it every time I say wave.


[User] Please transcribe this text

[Assistant] So what's the wave in this problem is that if I draw a plot of the height of the water,


[User] Please transcribe this text

[Assistant] versus the place where I dropped the rock. Maybe I should do a top view here.


[User] Please transcribe this text

[Assistant] This is the top view after a while you got these ripples emanating from this point. Or if you kept your eye next to the water, you will find these ups and downs going outwards.


[User] Please transcribe this text

[Assistant] So the wave is some displacement of a medium. That's what people say. That's not a perfect definition because we know electromagnetic waves have no medium. They travel in vacuum. But I think for this purpose, in this course, you should imagine a wave as what happens when you excite a medium.


[User] Please transcribe this text

[Assistant] So here's what, here's a way to think about the wave. You all understand the harmonic oscillator pretty well, right? Mass and spring system in equilibrium will sit there forever.


[User] Please transcribe this text

[Assistant] If you give it a kick, it'll start vibrating around its equilibrium position. There you have to keep track of only one variable, x, which is the location of the mass.


[User] Please transcribe this text

[Assistant] Of a medium, that means at every point in space there is something that's ready to oscillate.


[User] Please transcribe this text

[Assistant] So you start a system with one degree of freedom, system with infinite number of degrees of freedom, because the height of the water at each point is an independent variable. It can jump up and down with time. And that's what is oscillating.


[User] Please transcribe this text

[Assistant] So, if you don't do anything to the water, it'll stay at that height. But if you fiddle with it, you drop something, you hit it with something, it'll start vibrating.


[User] Please transcribe this text

[Assistant] But accept this vibration not of one variable, but of a variable in space and time.


[User] Please transcribe this text

[Assistant] x was a function of time and the name we use for the wave if something is oscillating such as the water.


[User] Please transcribe this text

[Assistant] Measuring from some point x, I'm going to use the symbol psi of x to denote the height of the water from its normal position at the point x at time t.


[User] Please transcribe this text

[Assistant] So x is not a dynamical variable here. x labels the point. Psi is the dynamical variable. Psi is the thing jumping up and down.


[User] Please transcribe this text

[Assistant] Okay, now, there are two kinds of waves. There's the longitudinal wave and the transverse wave.


[User] Please transcribe this text

[Assistant] And we define them as follows. The wave travels out with some velocity.


[User] Please transcribe this text

[Assistant] I'm going to use the symbol v for the velocity of the wave.