Now one may suspect that this piece is of little value except to some small band of specialists. I would humbly beg to differ, it is not a discourse on mathematics but on a change in how we are being told to think. Yes, we are being told to accept AI and its great "powers" and take as an answer whatever comes forth. After all these social media platforms use AI and look how wonderful they have become, NOT.
So the driver was a Technology Review piece, that MIT ersatz related rag that ofttimes yields insight into the minds of the maddening crowd of the social media generation. TR notes:
Unless you’re a physicist or an engineer, there really isn’t much reason for you to know about partial differential equations. I know. After years of poring over them in undergrad while studying mechanical engineering, I’ve never used them since in the real world. But partial differential equations, or PDEs, are also kind of magical. They’re a category of math equations that are really good at describing change over space and time, and thus very handy for describing the physical phenomena in our universe. They can be used to model everything from planetary orbits to plate tectonics to the air turbulence that disturbs a flight, which in turn allows us to do practical things like predict seismic activity and design safe planes. The catch is PDEs are notoriously hard to solve. And here, the meaning of “solve” is perhaps best illustrated by an example. Say you are trying to simulate air turbulence to test a new plane design. There is a known PDE called Navier-Stokes that is used to describe the motion of any fluid. “Solving” Navier-Stokes allows you to take a snapshot of the air’s motion (a.k.a. wind conditions) at any point in time and model how it will continue to move, or how it was moving before.These calculations are highly complex and computationally intensive, which is why disciplines that use a lot of PDEs often rely on supercomputers to do the math.
Then to the rescue comes AI, like a flash, we get the answer. AI excels where Matlab fails and humans, well we just do not count.
Now I tend to disagree. A good user of PDEs does so by "feeling" the answer, like Luke Skywalker. One can be experienced enough to see the boundary conditions, the infinite time results, the PDEs we have already solved. PDEs are used in Quantum mechanics, heat transfer, electromagnetic theory, and even cancer metastasis. We live in a spatio-temporal world definable in PDEs.
Take the simple one:
Time rate of change of X = Increase rate of X + Flow rate of X + Diffusion rate of X
Namely this thing X changes by means of it replicating, moving in some flow in and out, and by the random process of diffusing due to bouncing into one another. Most of the world as we know it goes that way. We do not need some large compute with AI to tell us this. We start by understanding if we have captured all the driving effects. Then we measure them again and again. We try to see if each of these effects is linear or some complex non-linear form. We remember the world is always linearizable in the small and this gives us a chance to get a small model. Then back to data gathering. You get it, it is incremental and we are always checking with Mother Nature.
This is the problem with Climate Change. It becomes a massive set of PDEs and we get into a never ending loop of checking.
But, and this is the critical BUT, we often find that our models, and that is what they are, have been delimited by some non-linearity. These factors result in massive changes and make our conclusions suspect. Using AI we may never see these factors, miss these phenomena, and thus diverge to some relatively false solution.
To be successful with PDEs one must live with them. Understand them. They are like predator animals, but after a while you find them like a pet, but still keep it at a distance. Placing it in some AI cage with darkened walls makes this creature foreign and you never really get to know it.
Thus my suggestion, go back to paper and pencil.
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