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Given a functional of the form

$$ J[y] = \int_{a}^b F(x,y,y')dx $$

the EL equation is given by

$$ \frac{dJ}{dy} = \left( \frac{\partial}{\partial y} - \frac{d}{dx}\frac{\partial}{\partial y'} \right) F $$

In applications however if often see the following rule for gradient descent

$$ \frac{\partial}{\partial t} y = -\left( \frac{\partial}{\partial y} - \frac{d}{dx}\frac{\partial}{\partial y'} \right) F $$

The questions isn't much about the formula per se, but how do we formalize the gradient descent in such case? I can't manage to find a proof or a derivation of the rule.

user8469759
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  • Out of curiosity, can you mention one of such applications? – caverac Jul 30 '18 at 12:47
  • Noise removal in image processing. Like this guy: https://math.stackexchange.com/questions/29703/euler-lagrange-gradient-descent-heat-equation-and-image-denoising?rq=1 – user8469759 Jul 30 '18 at 12:48
  • Doesn't the answer in that post solve your question as well? – caverac Jul 30 '18 at 13:03
  • No, because it seems to me is too short to get a proper insight. I don't know if for the name there's a method that might make it easier to look up. – user8469759 Jul 30 '18 at 13:06

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