TL;DR: what's the fastest way to transport a large amount of data over a mesh of many small pipes
As someone who knows little about this problem or the laws of physics, I wonder if it could be solved using physics -- set up a series of physical pipes, pump water from A to B, measure the flow at each point? If that would work, how complicated would the mesh need to be that solving it with physics is faster than brute-force computation?
You can directly model the flow of water through pipe networks computationally, the same way any chemical engineer would.
FWIW, as someone who designs massively parallel/distributed algorithms and went to school for chemical engineering, the conceptual model I use to design the aggregate behavior of complex networks are complex continuous flow chemical processes. All of the back pressure, flow overhead, reaction rates, equilibria, etc constructs are directly analogous to moving bits and doing computation in complex, distributed, heterogeneous computing systems. I think it is particularly effective for reasoning about distributed systems because chemical processes have no real concept of a global clock or shared state but still robustly and efficiently produce the desired output. It teaches you to reason about constructing optimal and robust global behaviors from subprocesses that only have local visibility and no explicit coordination between processes.
The fine article states that the authors had a breakthrough in 2011 based on modelling the system as a network of electrical resistances and obtaining the simultaneous currents rather than loading one edge at a time.
This would be an example of an analogue simulation. You first need to prove that the behavior of the physical analogue is also a solution to the original problem. The classic example is solving Poisson's equation, which gives (among other things) the electric field in a space containing charges, by measuring the deformation of an elastic sheet.
TL;DR: what's the fastest way to transport a large amount of data over a mesh of many small pipes
As someone who knows little about this problem or the laws of physics, I wonder if it could be solved using physics -- set up a series of physical pipes, pump water from A to B, measure the flow at each point? If that would work, how complicated would the mesh need to be that solving it with physics is faster than brute-force computation?