The Anatomy of a Dynamical System

hi everyone welcome back I'm Steve Brenton and I'm  really excited today to tell you about dynamical   systems and this lecture on the anatomy of a  dynamical system how you build them how you   analyze them how you understand them I absolutely  love dynamical systems they are our models for the   evolving world around us so they describe the rich  behavior of mixing fluids they're how we build   models that we would use to design rockets and  land them we use dynamical systems to understand   the brain disease networks social networks you  name it dynamical systems describe the evolving   world around us and this is really why I went  into applied math and science and engineering in   the first place is to model these rich systems  that change and that we interact with these   dynamical systems so this is going to be a video  on the anatomy of a dynamical system what are the   challenges of modern dynamical systems to describe  these really complex systems of interest and what   are some of the high-level objectives we have when  we're working with dynamical systems so I hope you   enjoy this this is oftentimes kind of an intro  piece of a longer lecture on how you would use   machine learning to learn these dynamical systems  or to learn how to control these systems but I   think this can be kind of a standalone little  piece on understanding what are the the integral   parts the bits and pieces that go together  to make up a dynamical system so dynamical   systems are our models of reality they are how  we describe the world around us especially how   things change and Co evolve in time so dynamical  systems are fundamentally linked to time and how   systems change in time okay so this is our kind  of simple model of a dynamical system it's a   system of coupled differential equations X is a  vector that represents the state of our system   it is a vector that has the minimal number set  of values that you need to describe your system   so for example if I have a pen then that pendulum  is uniquely defined not just by its angle-theta   but by its angular rate theta dot so the state  of that pendulum system is theta and theta dot   in a vector okay so that's what X is X is a vector  of the state of your system if I'm looking at the   stock market the state of the system might be  the value in all of the stocks and there might   be more information that needs to go into that  state vector if I'm looking at the weather it   might be the weather condition on a one kilometer  grid on planet Earth okay so the state vector X   can get very very large but it's a unique minimal  description of the system you care about okay F is   what we call the dynamics it's a set of functions  F 1 F 2 F 3 that describe the dynamics of state   1 state 2 state 3 so the time derivative of x1  will be given by the first row equation F 1 and   the time dynamics of x2 are given by F 2 and so  on and so forth so F is a vector valued function   that tells me given a current state X how does  that state change in the next time instant okay   so that is the dynamics it's a set of functions  oftentimes this is given by like Newton's second   law or something like that and we sometimes say  that this is a vector field because for every   point in space X this vector field essentially  tells you how the derivative is pointing and   what vector Direction X dot is pointing at that  point in space so it's a field of these Direction   vectors T is time and this is really important  because in lots of dynamical systems the system   itself changes in time ok this is super important  think about your brain state your brain state that   the dynamics the equations that would describe  how your brain works are fundamentally different   when you're sitting here watching a video lecture  versus when you're asleep different dynamics ok   so this is time varying the weather might change  the climate might change maybe we are actively   changing the dynamics of a system okay the ocean  dynamics the vector field of the ocean changes in   time good you are all of the variables that we  have active control over so maybe we have some   variables we can manipulate to try to change the  behavior of the system those are our control knobs   and those go in this vector U so in the case of  the pendulum maybe I put that pendulum on a cart   and I can accelerate that cart I can drive that  cart around and that would be my control and put   you to try to change the dynamics of my system  beta are all the other parameters of the system   that I don't explicitly have control over but I  might want to analyze and understand so in the   pendulum example this might be like the length of  the pendulum arm or the mass of the pendulum and   things like that and so these beta parameters  we want to understand the explicit dependence   of these dynamics on those beta parameters as I  change those parameters I might get really big   changes in the dynamics called bifurcations and  so these beta parameters are super important good   so that's our dynamical system now in addition  we also have measurements of our systems so we   don't always have access to this full state X  because maybe my system is exceedingly large   you can't measure all of the brain all of the  neurons in your brain that would be you know   completely impractical so maybe you can measure  a really really small subset or even some crude   measurement some proxy measurement like a EKG  or EEG or something like that so you have these   measurements in Y and I now that we've kind of  talked about what is the anatomy of a dynamical   system I want to walk you through one of the  modern challenges so people have been writing   down dynamical systems and studying them and  analyzing them for centuries right Newton's law   second law is a dynamical system that was kind of  the beginning of our foray into dynamical systems   but in the modern era a lot has changed and there  are some big challenges that were able to start   tackling now okay and so that's that's really what  I want to talk about here so in modern dynamical   systems some of the biggest challenges probably  the biggest one that goes unmentioned in a lot   of cases is that in most situations we actually  don't know we don't know the dynamics we don't   have a model of our system so for mechanical  systems like the pendulum yes you can derive   you know Newton's laws or the Lagrangian or the  Hamiltonian and you can derive these equations of   motion from first principles that's like you crack  your physics book and you derive this system from   scratch that's how we you know used to do things  that's how we do things when we can but for modern   systems of interest we don't have access to  these governing equations and F there is no   known master equation for how your brain works  we don't know the F for a brain we don't know   the F for a disease Network we have models we have  approximations but we can't agree on one governing   physical law for how a disease spreads or how the  climate works ok so modern systems of interest F   is often unknown even in systems where you do know  F like the Navy or stokes equations for a fluid   it might be far too complicated to use and so we  would need to learn a reduced order model in terms   of the things we care about in terms of the less  descriptive state X that captures the big things   I care about in my fluid the other really really  big challenge in dynamical systems and I think   this this one is really maybe tied for number one  is non-linearity so this function f in general for   systems of interest is often nonlinear and that  means superposition of solutions doesn't hold if   my dynamics were linear if X dot equals a matrix  a times X everything would be simple there would   be a canonical closed form solution you could  just write down the solution and go home we would   all be out of business so non-linearity is what  gives us job security and dynamical systems this   is the real challenge in many systems is that  even small amounts of non-linearity make it so   there are not closed form solutions in general  there are special classes of nonlinear systems   and special solutions that you can find but there  is no generic one size fits all nonlinear theory   of nonlinear dynamical systems so that's really  interesting and most systems we care about are   nonlinear other challenges are dimensionality of  the state vector X so for many many systems this   state vector X might be exceedingly complex I  think in your brain there are estimates that   you have a hundred billion neurons that's a  hundred billion dimensional state vector just   to describe the state of all of those neurons  uniquely describing the climate or the weather   is also very very massive problem if you want to  get a really good description of weather on planet   earth you need a very high resolution and Earth's  pretty big ok disease modeling if you want to   actually do a good job of this you really need to  figure out like the network of how people interact   and that's you know there's billions of us and  there's tons of connections in this network so   it's super complicated high dimensional dynamics  in a lot of cases and that makes it really hard to   analyze these along with dimensionality is multi  scale so these dynamical systems are not just big   but there's a range of scales from you know big  scales all the way to really really small scales   so think about whether or or climate you know on  the scale of the size of Earth there are things I   care about their seasonal changes you know big  hurricanes but little teeny-tiny features also   might matter for the weather prediction three  weeks out ok so you have this range of scales   you care about your brain also is fundamentally  multi scale spatially there's regions and then   those regions are composed of you know individual  neurons and things but there's also multi scale   in time you're listening to me talk about this  right now you're hearing words I'm saying on   seconds you're connecting that to things I  told you a few minutes ago but you also have   this lifetime of experience over months and  years so you're extremely multi scale in time   dynamical systems are fundamentally chaotic in a  lot of situations so not all nonlinear systems are   chaotic but many of them are and what that means  is that you get a very sensitive dependence of   the outcome of these systems you get a very the  future depends very strongly on small changes on   the initial condition and the parameters and the  dynamics now so if you have your system and you   change your parameter even by a little bit you  might get very very different phenomena or if   you change your initial condition a little bit you  might go to very different places and that makes   it really hard to predict these systems so that's  actually why weather is so hard to predict you   know we can't predict three weeks out because this  is a super chaotic system and we can only measure   the system so accurately and we only can describe  the equations so accurately and the parameter so   accurately so chaos is is still a huge problem  latent variables hidden variables again this is   related to dimensionality if I have a big state  vector chances are I'm not measuring everything   you can't measure all of the neurons in your  brain you might only measure a few of them you   might only have a proxy measurement so latent  variables hidden variables is a big deal they   might matter a lot and you can't measure them what  do you do in those situations all real dynamical   systems have disturbances and noise and you have  to consider this so your dynamical system is going   to be forced by external disturbances stochastic  forcing and your measurements are going to have   noise and you can't always just assume that these  are Gaussian white noise processes like what we do   in textbooks because that's the only thing that  we can actually solve in the real world you're   probably going to have correlated structured  biased noise and disturbance terms and All of this gets wrapped up into this concept of uncertainty.  There's a huge amount of uncertainty in every   single piece of this dynamical system. I have  uncertainty in the model 'f'. I have uncertainty in   the parameters beta. I have uncertainty in what I'm  measuring and how it's being measured. I might even   have uncertainty in the measurement time of when  I take that measurement. I mean there's all kinds   of uncertainty and that all will conspire  to make this a very challenging problem. I have asterisks on the challenges because I want  to point out that most, maybe not all, but most   of these problems can be recast as optimization  problems. So optimally handling noise and   disturbances for future predictions or estimation.  Optimally propagating uncertainty in the future...   And so on and so forth. Optimally reconstructing  the full state from limited measurements 'y'. And when I say that this is an optimization problem  you should be thinking now "I can probably start   using tools from machine learning" because  machine learning is just optimization based on data it's building models using optimization  and a wealth of data and so this is really nice   because modern dynamical systems kind of the  anatomy of a dynamical system allows you to focus   on individual tasks and challenges with these  emerging optimization techniques. So now that we've understood kind of the anatomy of  this dynamical system I want to talk a little bit   about how we would use it and there's so many  uses I'm just gonna you know briefly   touch on some of them of course we might want to  predict the future state of the system so I might   want to predict the weather a week from now when  I think of future state prediction I think of you   know what if someone's lost at sea okay so we had  some rough idea of their initial condition and we   want to integrate where we think they'll be two  hours later okay so that's a future prediction   and the future prediction doesn't have to be  deterministic it might be statistical I might   have to run an ensemble of initial conditions  maybe I have uncertainty of where that person   was lost at sea or maybe this is the Gulf oil  spill and I don't know exactly where the oil is   coming up so I have a Gaussian of uncertainty and  I want to propagate that into a future estimate of   where the distribution of oil will be so that's  a statistical or an ensemble prediction and I'll   point out all of these topics every term in this  dynamical system every one of those challenges   every one of these uses there are dozens if not  hundreds of people diligently working on these   problems these are not challenges these are whole  careers okay design optimization that's another   big use of dynamical systems models what I might  want to do is now that I have a model of my system   I might want to design betas to get a performance  or an output that is desirable so think Formula   one cars or super yachts or rocket engines I want  to design the parameters of my system to get some   desired high-level output like drag and lift and  things like that okay and so that requires you to   be able to understand how your system depends on  these parameters now if you do that optimization   in real time based on your measurements that  would be called feedback control so if you can   actively manipulate some part of your system in  real time those parameters become actuators and   this becomes a feedback control problem and that  would be really satisfying and incredibly useful   in many many systems if we can actively control  them to some engineering you know specifications   now most of the the first uses are very practical  very engineering oriented but we also as humans   deeply care about understanding the world  around us we don't just want to manipulate   it we don't just want to predict it we want to  understand fundamentally how and why it works   okay and these dynamical systems give you a  very kind of intuitive way of understanding   how the world evolves and what are the rules for  how that evolves and so there's a couple of kind   of key words here that I think are really  important interpretable and generalizable   so interpretable means that this model is not so  complex that I can't communicate with you about   it we could if this model is not too complicated  we can talk about it we can try to understand it   we can communicate and interpret this model  generalizable is also really important if I   write down this model I want it to be useful in  other scenarios that I might not have thought   of so being you know having some generality and  how it how this model works even for parameters   I haven't tested that's really important and I  always think of Newton's second law is kind of   the ultimate interpretable and generalizable  dynamical systems model F equals MA it's super   simple so we can interpret and understand it it is  very generalizable it works in so many scenarios   you know for different parameters for different  state vectors and so on and it fundamentally has   changed the way that we understand with and  interact with the world so we understand the   world better because of that interpretable model  okay good so that's kind of the overview of the   anatomy of a dynamical system I strongly encourage  you to start playing around with systems yourself   pick a system that you're interested in the brain  or an epidemiological system or the climate pick   something interesting and start thinking  to yourself what would be a state vector of   that system what would be the control inputs and  the parameters what might be some rules for how   that system evolves in time what can I measure  and start thinking about how those challenges   and these uses fit into your understanding  of your dynamical system alright thank you

Channel: Steve Brunton

Views: 32,482

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Id: -FvrON0OmYc

Length: 17min 53sec (1073 seconds)

Published: Fri Jul 30 2021