4 ms·

Most useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the m

Most useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the moment. See physics informed neural networks: https://en.m.wikipedia.org/wiki/Physics-informed_neural_networks https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...

marginalia_nu 5y ago
The point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.
- medo-bear 5y ago
  i disagree. equations are useful only if you have use for them. this is by definition of the term. on the other hand, high schoolers rarely feel like Hooke's law is useful!
- pvarangot 5y ago
  Isn't Hooke's law a solution to the harmonic motion PDE in that page?
  - tagoregrtst 5y ago
    No? Hooks law is an approximation to a material (and spring) property that sets the PDE up. Sin(x) is the solution. But I could be wrong :S
    - capn_duck 5y ago
      ASin(wx) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.
      - tagoregrtst 5y ago
        I meant its a solution to a specific problem set up using Hooke’s law
    - kergonath 5y ago
      Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator. It’s not surprising if you see a solid material that way: - it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum - therefore either extending it or compressing it increases its energy - therefore the second derivative of its energy is positive. On a high level, this sounds very much like a simple parabola, because it is one. For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator. Alternatively, to a physicist, almost everything looks like a parabola. If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law). The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well. I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).
  - throwaway81523 5y ago
    Harmonic motion on that page is an ODE rather than a PDE. The whole page is kind of strange since it doesn't explain much.
- aqme28 5y ago
  I think we need a better mutual definition of useful. IMO an equation is "useful" if someone out there is doing practical things with it. You are taking a sort of definition where anyone can do math with it.
  - squeaky-clean 5y ago
    Well the page title is "PDEs You Should Know", not "PDEs Someone Should Know"
- aqme28 5y ago
  The math isn't that hairy. I took numerical methods and computational physics classes in undergrad.
  - deleted 5y ago
    [deleted]
- blablabla123 5y ago
  I think the most generic approach is to prove existence and uniqueness and then go with numerical methods.
- nautilius 5y ago
  You really should read up on numerical math, we really don't need closed form solutions, and this makes PDE far from useless.
- blueplanet200 5y ago
  Depends on what you're doing, but you can numerically simulate solutions to PDEs. To me that makes them useful even if we struggle to conjure up closed form solutions.

marginalia_nu 5y ago

The point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.

medo-bear 5y ago
i disagree. equations are useful only if you have use for them. this is by definition of the term. on the other hand, high schoolers rarely feel like Hooke's law is useful!
pvarangot 5y ago
Isn't Hooke's law a solution to the harmonic motion PDE in that page?
- tagoregrtst 5y ago
  No? Hooks law is an approximation to a material (and spring) property that sets the PDE up. Sin(x) is the solution. But I could be wrong :S
  - capn_duck 5y ago
    ASin(wx) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.
    - tagoregrtst 5y ago
      I meant its a solution to a specific problem set up using Hooke’s law
  - kergonath 5y ago
    Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator. It’s not surprising if you see a solid material that way: - it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum - therefore either extending it or compressing it increases its energy - therefore the second derivative of its energy is positive. On a high level, this sounds very much like a simple parabola, because it is one. For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator. Alternatively, to a physicist, almost everything looks like a parabola. If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law). The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well. I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).
- throwaway81523 5y ago
  Harmonic motion on that page is an ODE rather than a PDE. The whole page is kind of strange since it doesn't explain much.
aqme28 5y ago
I think we need a better mutual definition of useful. IMO an equation is "useful" if someone out there is doing practical things with it. You are taking a sort of definition where anyone can do math with it.
- squeaky-clean 5y ago
  Well the page title is "PDEs You Should Know", not "PDEs Someone Should Know"
aqme28 5y ago
The math isn't that hairy. I took numerical methods and computational physics classes in undergrad.
- deleted 5y ago
  [deleted]
blablabla123 5y ago
I think the most generic approach is to prove existence and uniqueness and then go with numerical methods.
nautilius 5y ago
You really should read up on numerical math, we really don't need closed form solutions, and this makes PDE far from useless.
blueplanet200 5y ago
Depends on what you're doing, but you can numerically simulate solutions to PDEs. To me that makes them useful even if we struggle to conjure up closed form solutions.