The Earth Movers Distance is a concept in probability theory and information theory. In short one can say it is a distance measure between two probability densities $p_1(x), p_2(x)$ so that probability is moved from one density to another at the cost $|x_2-x_1|d(x_1,x_2)$ if $d(x_1,x_2)\in\mathbb R^2\to \mathbb R^1$ is the total probability density moved from point $x_1$ to $x_2$. ( I am writing from memory and intuition here so please correct me if I am wrong: )
$$d = \min_{d}\left\{\int_{\mathbb R^2} |x_1-x_2| d(x_1,x_2) \right\} \, s.t. \, \cases{f_1(x_1) = \int d(x_1,x)dx\\f_2(x_2)=\int d(x,x_2)dx\\d(x_1,x_2)\geq 0\,\,\, \forall (x_1,x_2)}$$
So we are to find the function $d$ which minimize the above expression and also fulfills the constraints. If we are used to working with optimization we know that we can rewrite the first equality: $$0 = \int d(x_1,x)dx - f_1(x_1)$$ And turn it into a term in our optimization:
$$+\left\|\int d(x_1,x)dx - f_1(x_1)\right\|$$ For some suitable choice of norm.
I know of some discrete optimization methods and heuristics (for example the Hungarian algorithm) are being used to solve this problem, but somehow I feel it should be possible to solve with differential equations. Do you know of any methods which use differential equations and / or some energy minimization to find the Earth Movers Distance?
There has been quite a bit of of recent work on this problem; in seismic waveform inversion, see, for example, Brittany Froese and Bjorn Enquist - Application of the Wasserstein metric to seismic signals, where the authors cast the minimization of the earth-mover's distance as a problem in solving a Monge-Ampere equation.
Hope this helps,
Tom