3 ms·

> one of the classic problems of applied mathematics. Maybe I'm biased with my physics PhD, but I'd call it a physics problem. "Solving" it to me means a close

by spodek 7y ago

> one of the classic problems of applied mathematics. Maybe I'm biased with my physics PhD, but I'd call it a physics problem. "Solving" it to me means a closed-form solution, not a particular prediction, but I guess the applied math problem is a different problem with a different solution.

hpcjoe 7y ago
(another Physics Ph.D. here) Solving a problem in physics or math generally implies a closed form solution. Which this is, decidedly, not. It is in fact a special case, with low particle count, and doesn't return velocity information, thus leading to problems assessing whether some things like Energy, are actually conserved. They discuss this. And have to use a second network to compute velocity, used for computing kinetic energy. In which they discover that in close near-collision dynamics, their error can spike. Unless they reproject the phase space coordinates specifically atop the "right" constant energy surface. So ... yeah ... title of preprint is quite misleading. Their conclusions are also misleading. Are they better/more general at numeric solution for arbitrary mass, arbitrary particle count, arbitrary initial conditions than, say, a good symplectic integrator of high order? No. Actually they only studied the equal mass case of 3 particles. Given the combinatorics, I could not fathom the size of the training set they'd need for N(particles) of any reasonable size. There they would start having to look into some of the approximations that the simulation folks have been using for a long ... long time. Just to get their training cost down far enough to be useful. An example. Take a simulation with N ~ 10<sup>6</sup>. 1 Million objects. Take a mass distribution in a Poissonian distribution, or an EVD, or similar. Make it similar to approximate stellar masses. Create your initial conditions and generate your training sets from these initial conditions. How many of these simulations would you need to run, and for how long, in order to be able to get something which provided long time scale prediction, which meshed with reality? Oh, take N at least 8 for planetary mass distribution. With satellites, so N ~ 200. Never mind planetary rings or Oort cloud/Kuiper belt/asteroid belt objects. How does large does your training need to grow in order to be able to predict, with any degree of accuracy, orbital stability over gigayear intervals? My point is not to beat on the authors, but to point out that this study, while interesting, is most definitely not a solution to the 3 body problem. There is vast literature on solutions to N-body problems in general. These problems are considered hard. Because, they actually are. There are some "short cuts", basically field-like approximations, that one can use to make some generally reasonable arguments about the interaction with many distant objects. They actual dynamics, and details of Poincare maps and phase space portraits are, again, very well studied phenomenon. With 100+ years of research behind this. If they wanted a simpler dynamic problem to work with, the double pendulum exhibits chaotic behaviour. All they need to vary are 2 parameters for each arm/mass, length and mass. See how close they come to actual behavior, onset of trajectory divergence, etc. [edit: slight clarification of second network computing velocity for kinetic energy]
mikorym 7y ago
How is it different? Edit: Sorry, I assume you refer to the OP in which case yes, I agree that it's not a solution at all. Rather, it is a presumably faster-than-before simulation.
oefrha 7y ago
Physics PhD here. I'd call it both. A lot of what applied math departments study are problems of physical nature; and a lot of physics research boils down to applied math. (Broadly speaking, all theoretical physics is applied math.) Guess I'm biased too because I'm a theorist and started out from a math background.
- ska 7y ago
  I went the other direction, physics to math, but agree with your broad characterization. Not all applied math is physics, obviously. I do find mathematicians and physicists often have different characteristic ways of thinking about similar things, but the lines are blurry.

by spodek 7y ago

hpcjoe 7y ago

(another Physics Ph.D. here) Solving a problem in physics or math generally implies a closed form solution. Which this is, decidedly, not. It is in fact a special case, with low particle count, and doesn't return velocity information, thus leading to problems assessing whether some things like Energy, are actually conserved. They discuss this. And have to use a second network to compute velocity, used for computing kinetic energy. In which they discover that in close near-collision dynamics, their error can spike. Unless they reproject the phase space coordinates specifically atop the "right" constant energy surface. So ... yeah ... title of preprint is quite misleading. Their conclusions are also misleading. Are they better/more general at numeric solution for arbitrary mass, arbitrary particle count, arbitrary initial conditions than, say, a good symplectic integrator of high order? No. Actually they only studied the equal mass case of 3 particles. Given the combinatorics, I could not fathom the size of the training set they'd need for N(particles) of any reasonable size. There they would start having to look into some of the approximations that the simulation folks have been using for a long ... long time. Just to get their training cost down far enough to be useful. An example. Take a simulation with N ~ 10<sup>6</sup>. 1 Million objects. Take a mass distribution in a Poissonian distribution, or an EVD, or similar. Make it similar to approximate stellar masses. Create your initial conditions and generate your training sets from these initial conditions. How many of these simulations would you need to run, and for how long, in order to be able to get something which provided long time scale prediction, which meshed with reality? Oh, take N at least 8 for planetary mass distribution. With satellites, so N ~ 200. Never mind planetary rings or Oort cloud/Kuiper belt/asteroid belt objects. How does large does your training need to grow in order to be able to predict, with any degree of accuracy, orbital stability over gigayear intervals? My point is not to beat on the authors, but to point out that this study, while interesting, is most definitely not a solution to the 3 body problem. There is vast literature on solutions to N-body problems in general. These problems are considered hard. Because, they actually are. There are some "short cuts", basically field-like approximations, that one can use to make some generally reasonable arguments about the interaction with many distant objects. They actual dynamics, and details of Poincare maps and phase space portraits are, again, very well studied phenomenon. With 100+ years of research behind this. If they wanted a simpler dynamic problem to work with, the double pendulum exhibits chaotic behaviour. All they need to vary are 2 parameters for each arm/mass, length and mass. See how close they come to actual behavior, onset of trajectory divergence, etc. [edit: slight clarification of second network computing velocity for kinetic energy]

mikorym 7y ago

How is it different? Edit: Sorry, I assume you refer to the OP in which case yes, I agree that it's not a solution at all. Rather, it is a presumably faster-than-before simulation.

oefrha 7y ago

Physics PhD here. I'd call it both. A lot of what applied math departments study are problems of physical nature; and a lot of physics research boils down to applied math. (Broadly speaking, all theoretical physics is applied math.) Guess I'm biased too because I'm a theorist and started out from a math background.

ska 7y ago
I went the other direction, physics to math, but agree with your broad characterization. Not all applied math is physics, obviously. I do find mathematicians and physicists often have different characteristic ways of thinking about similar things, but the lines are blurry.