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Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations

54 pointsby takiwatangaabout 3 years ago

7 comments

btrettelabout 3 years ago
Fluid dynamicist here. I think these sort of blow-up results get far too much attention from the popular science press, and I don&#x27;t know why. This is <i>practically</i> a pure math problem. Both the Euler and (constant viscosity) Navier-Stokes equations are known to be flawed models in certain situations. And corrections for basically any identified flaw exist. Just because a certain category of flaw hasn&#x27;t been proved yet doesn&#x27;t mean much; its correction might already be published!<p>From what I understand, in the 1930s Jean Leray provided that the 2D (constant viscosity) Navier Stokes have smooth and unique solutions. And in the 1960s Olga Ladyzhenskaya proved that for certain non-constant viscosity laws (which are a lot more realistic than constant viscosity), the 3D Navier Stokes equations have smooth and unique solutions. From a physics perspective, there is no mystery here in terms of whether a properly posed problem will blow-up.<p>A lot of people (for example: [0]) link the question of whether the Navier-Stokes equations have smooth and unique solutions to turbulence, which makes little sense to me. Take the situations mentioned in the previous paragraph where proofs exist. The turbulence problem still exists there! The problem of turbulence is one of computational complexity. Turbulent flows seem to require a lot of computation.<p>[0] <a href="https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;the-trouble-with-turbulence-20190128&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;the-trouble-with-turbulence-2...</a>
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supernova87aabout 3 years ago
I&#x27;ll believe that deep learning has really achieved human level performance when the algorithm looks at the Navier-Stokes equations and says, &quot;eh, it&#x27;s not like I&#x27;m ever actually going to use that after grad school&quot;.
sweezyjeezyabout 3 years ago
Mathematics demigod Terry Tao came up with an extremely cool idea for how to prove blow-up results like this. <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=DgmuGqeRTto" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=DgmuGqeRTto</a>. The idea is to try and make a self-replicating &quot;fluid computer&quot; - think of our imaginary fluid as a continuous set of points which are all moving independently. If you set things up in very particular way, you can treat patches of fluid like components in a machine - these interact with each other, moving energy around to different parts of the fluid, until eventually you end up with a new copy of the machine - but smaller - and with more average velocity than the original. This then repeats, smaller and smaller, faster and faster, until within a finite time you have a singularity. He used this to prove blow up in some &#x27;toy&#x27; versions of the Euler equations, with extra rules &#x2F; parameters to make it work, but he does (or did) seem to think this has some chance of working in the real model.<p>(disclaimer - I do not have expertise in this field)
smitty1eabout 3 years ago
Great link. Hats off to the writers, as well as the researchers. Coming up with these ideas, implementing them, and them communicating them to a non-specialist audience in an engaging way takes serious effort. Thank you.
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pclmulqdqabout 3 years ago
This is a really interesting application of DL, and something that it should be well-positioned to do. I&#x27;m excited for the results!
westurnerabout 3 years ago
Euler equations (fluid dynamics) <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Euler_equations_(fluid_dynamics)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Euler_equations_(fluid_dynamic...</a><p>&gt; <i>In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler.</i> In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. [1]<p>Navier–Stokes equations <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Navier%E2%80%93Stokes_equations" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Navier%E2%80%93Stokes_equation...</a><p>&gt; <i>The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass </i> for Newtonian fluids.<p>Computational fluid dynamics <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Computational_fluid_dynamics" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Computational_fluid_dynamics</a><p>Numerical methods in fluid mechanics <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Numerical_methods_in_fluid_mechanics" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Numerical_methods_in_fluid_mec...</a><p>- [ ] Quantum fluid <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quantum_fluid" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quantum_fluid</a><p>&gt; <i>Quantum mechanical effects become significant for physics in the range of the de Broglie wavelength. For condensed matter, this is when the de Broglie wavelength of a particle is greater than the spacing between the particles in the</i> lattice <i>that comprises the matter.</i> [...]<p>&gt; <i>The above temperature limit T has different meaning depending on the quantum statistics followed by each system, but generally refers to the point at which the system manifests quantum fluid properties. For a system of fermions, T is an estimation of the Fermi energy of the system, where processes important to phenomena such as superconductivity take place. For bosons, T gives an estimation of the Bose-Einstein condensation temperature.</i><p>Classical fluid <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Classical_fluid" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Classical_fluid</a><p>&gt; <i>Classical fluids [1] are systems of particles which retain</i> a definite volume, <i>and are at sufficiently high temperatures (compared to their Fermi energy) that quantum effects can be neglected</i> [...] <i>Common liquids, e.g., liquid air, gasoline etc., are essentially mixtures of classical fluids. Electrolytes, molten salts, salts dissolved in water, are classical charged fluids. A classical fluid when cooled undergoes a freezing transition. On heating it undergoes an evaporation transition and becomes a classical gas that obeys Boltzmann statistics.</i><p>Chaos theory <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Chaos_theory" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Chaos_theory</a><p>&gt; <i>Chaos theory is [...] focused on underlying patterns and deterministic laws highly sensitive to initial conditions in dynamical systems that were thought to have completely random states of disorder and irregularities. [1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.</i><p>Quantum chaos <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quantum_chaos" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quantum_chaos</a><p>&gt; <i>If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?</i><p>... in dynamic, nonlinear - possibly adaptive - complex systems.
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jiehongabout 3 years ago
Is it only me, or their menu bar is hovering on the text in a broken way on iOS Safari[0]?<p>I remember seeing that buggy behaviour a few months ago, but as it’s still there, I’m starting to think not everyone might see it.<p>[0]: <a href="https:&#x2F;&#x2F;ibb.co&#x2F;x1ngmk1" rel="nofollow">https:&#x2F;&#x2F;ibb.co&#x2F;x1ngmk1</a>
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