For some reason or the other, this article took me mentally to a place of familiarity and comfort, of having “been there, done that”. This happened even though I have no experience with the subject of fluid dynamics – beyond perhaps the faint memories of a course or two taken during my undergraduate studies. The phrase “Navier-Stokes equations” sounds vaguely familiar. And I do have friends who worked on fluid dynamics for their doctoral theses.
I think that the way my brain is wired, I have tended to approach problem-solving in the past in general with a similar overall philosophy of attack as that which comes through in the article. One should be adaptable in the solution approach. One should not be afraid of patching together unconventional and unorthodox approaches for problem solving. Blindly following formulae can sometimes only get you so far. I have sometimes thought my thought process when visualizing problems and solutions did not necessarily belong in the mainstream, but perhaps that is not completely true these days.
However, having said all this, one should not make the mistake of underestimating the utility of characterizing processes using standard approaches, including the use of equations (or formulae) as applicable, to simplify problem solving. Many of our technological advances would not have been possible without recognizing a structure behind certain phenomena or observations, often stated in the form of equations. For example, without the use of Newton’s Laws of Motion, and the equations that follow, we would not be able to shoot and direct objects to the distant recesses of Space – and direct these objects precisely to where we want them to be at particular time instants – to be able to learn more about our place in the scheme of Everything. We used computers initially to speed up our use of these equations. Even during my time we could be innovative about the the use of computers for the application of these equations. I could actually come up with evaluation and analysis tools using techniques and concepts like iteration, convergence, equilibrium, intersection points, even educated guesses for solution spaces, etc.., based on standard equations that could be implemented on computers – which could lead to solution spaces that would not easily be accessible by hand. But we are now getting past even that stage. We are making use of the power of the computers in other ways for discovering and intuiting the structures themselves, and then using these structures to further problem solving.
As time passes, we also learn that many of the equations that we depend upon in our daily lives may be approximations of what is actually physically going on in the Universe. We still have not connected all the four “fundamental” forces of nature that we recognize today in a logical way. The relevant equations that characterize the individual forces do not connect with each other. There is something missing. Something else is going on.
As another aside, how many of us were aware that the GPS system that we have come to depend upon to find our locations on Earth would be inaccurate without accounting for the physical impact of the Theory of Relativity. Yes, the speed of light does have an impact in a practical sense in our lives even if we are not aware of it, and Newton’s equations by themselves are not adequate for some applications even though they would suffice for others. Pause for a minute to ponder the fact that Euclidean geometry, a tool that seems to work so well for us in characterizing our physical world, is not the only kind of geometry that exists, and that there are other kinds of geometries that obey different rules that can be useful for studying and understanding different systems and applications, and perspectives.
Coming back to simpler things like fluid flows, the subject of the attached article, there have been equations that have formed the basis of our understanding of fluid dynamics for years. These equations help us predict behaviors and design systems. But then we also learn from the article that there may be singularities, and the possible existence of a place or space where the equations do not quite add up (pun intended). The equations can actually break down. What do you do when the equations blow up, perhaps as one or more of the parameters tends towards zero and the solution tends to infinity? Do we need to push ourselves to try provide structure, perhaps in the form of equations, to our understanding everywhere, including at perceived or real singularities? In my mind, it only makes sense to the extent that these formulations help us to predict and foresee how real physical (and other) phenomena will play out in these “extreme” circumstances, thereby expanding our knowledge of the world, or creating the possibility of some practical application that may serve our benefit. I admit that I myself have a curiosity about what lies beyond the singularity of a Black Hole.
Enter now the new techniques and concepts like artificial intelligence, deep learning, neural networks, etc.., aided and abetted by the availability of massive data collection and storage capability, massive processing power, and even the emergence of innovative technologies like quantum computing. Essentially, you can gather up all your observations and process the data using your computer until you are at point where you can try to see a pattern (or structure). It may be a pattern that is easy to pick up on, or it may be something that is weak and difficult to follow. So you continue the computing process for as long as is necessary. With the current state of our technology, this process may be easier than trying to mentally discern a a structure or process that lies hidden, something that is not obvious to the common eye. After all, how many of us have the intuition of an Issac Newton to get to figure out concepts like Gravity?
The general approach described in the article below seems to me to be a combination of multiple techniques in what could be considered by some to be a somewhat unorthodox manner. The approach follows a methodology and a way of thinking that I actually feel quite comfortable with. The approach seems quite intuitive to me, and its usage would come naturally to me. Although I have no experience with the deeper science of artificial intelligence and deep learning, I would be quite happy making use of these tools as applicable and available to further whatever research or analysis that I might happen to find myself involved in. I would instinctively tend to look at the whole problem holistically without necessarily constraining my thinking to some misguided notion to purity of procedure.
The question that still bugs me regarding the article below is that of the reason why significant brainpower and computer resources are being expended on this particular topic. Why are people so intent on “Blowing Up” the equations? Is it just intellectual curiosity (dare I call it idle curiosity), or is there some practical use that they have in mind today? Either way, I can see myself spending time implementing techniques and investigating phenomena just like this, even if it were just for the fun of it.
Deep Learning Poised to ‘Blow Up’ Famed Fluid Equations | Quanta MagazineDeep Learning Poised to ‘Blow Up’ Famed Fluid Equations | Quanta Magazine