# Stephen Smith's Blog

Musings on Machine Learning…

# Introduction

One of the arguments against Strong AI relates to how computers can somehow break out of their programming to be creative. Or how can you program a computer to be self-aware? The argument is usually along the lines that AIs are typically programmed with a lot of linear algebra (matrix operations) to form Neural Networks, or are programmed as lots of if statements like in Random Forests. These seem like very predetermined operations and how can they ever produce anything creative or beyond what it is initially trained to do?

This article is going to look at how fairly simply defined systems can produce remarkably complex behaviours that go way beyond what would be imagined. This study started with the mathematical analysis of how physical systems can start with very simple behaviour and as more energy is added their behaviour becomes more and more complex resulting in what appears to be pure chaotic behaviour. But these studies show there is a lot of structure in that chaos and that this structure is quite stable.

The arguments used against Strong AI, also apply to the human brain which consists of billions of fairly simple elements, namely our neurons that somehow each perform a fairly simple operation, yet combined yield our very complex human behaviour. This also can be used to explain the spectrum of intelligence as you go up the evolutionary chain from fairly simple animals to mammals to primates to humans.

# Taylor Couette Flow

Taylor Couette Flow is from fluid mechanics where you have an experiment of water between two cylinders. Fluid mechanics may seem far away from AI, but this is one of my favourite examples of the transition from simple to complex behaviour since it’s what I wrote my master’s thesis on long ago (plus there really is a certain inter-connectedness of all things).

Consider the outer cylinder stationary and the inner cylinder rotating.

At slow speeds the fluid close to the inner cylinder will move at the speed of that cylinder and the fluid next to the outer cylinder will be stationary. Then the fluid speed will be linear between the two to give nice simple non-turbulent flow. The motion of the fluid in this experiment is governed by the Navier-Stokes equations, which generally can’t be exactly solved, but in this case it can be shown that for slow speeds this is the solution and that this solution is unique and stable (to solve the equations you have to assume the cylinders are infinitely long to avoid end effects). Stable means that if you perturb the flow then it will return to this solution after a period of time (ie if you mix it up with a spoon, it will settle down again to this simple flow).

As you speed up the inner cylinder, at some point centrifugal force will become sufficient to cause fluid to flow outward from the inner cylinder and fluid to then flow inward to fill the gap. What is observed are called Taylor cells where the fluid forms what looks like cylinders of flow.

Again the Navier Stokes equations are solvable and we can show that now we have two new stable solutions (the second being the rotation is in the opposite direction) and that the original linear solution, although it still exists is no longer stable. We call this a bifurcation, where we vary a parameter and new solutions to the differential equations appear.

As we increase the speed of this inner cylinder, we will have further bifurcations where more and more smaller faster spinning Taylor cells appear and the previous solutions become unstable. But past a certain point the structure changes again and we start getting new phenomena, for instance waves appearing.

And as we keep progressing we get more and more complicated patterns appearing.

But an interesting property is that the overall macro-structure of these flow is stable, meaning if we stir it with a spoon, after it settles down it will appear the same at the macro level, indicating this isn’t total random chaotic behaviour but that there is a lot of high level structure to this very complicated fluid flow. It can be shown that often these stable macro-structures in fact have a fractal structure, in which case we call them strange attractors.

This behaviour is very common in differential equations and dynamical systems where you vary a parameter (in our case the speed of the inner cylinder).

If you are interested in some YouTube videos of Taylor Couette flow, have a look here or here.

# What Does This Have to Do with Intelligence?

OK, this is all very interesting, but what does it have to do with intelligence? The point is that the Taylor Couette experiment is a very simple physical system that can produce amazing complexity. Brains consist of billions of very simple neurons and computers consist of billions of very simple transistor logic gates. If a simple system like Taylor Couette flow can produce such complexity then what is the potential for complexity beyond our understanding in something as complicated as the brain or computers?

In the next article we’ll look at how we see this same behaviour of complexity out of simplicity in computer programs to start see how this can lead to intelligent behaviour.