*The problem is that there are orders of magnitude more mathematical functions than possible networks to approximate them. And yet deep neural networks somehow get the right answer.*

Now Lin and Tegmark say they’ve worked out why. The answer is that the universe is governed by a tiny subset of all possible functions. In other words, when the laws of physics are written down mathematically, they can all be described by functions that have a remarkable set of simple properties.

So deep neural networks don’t have to approximate any possible mathematical function, only a tiny subset of them.

To put this in perspective, consider the order of a polynomial function, which is the size of its highest exponent. So a quadratic equation like y=x2 has order 2, the equation y=x24 has order 24, and so on.

Obviously, the number of orders is infinite and yet only a tiny subset of polynomials appear in the laws of physics. “For reasons that are still not fully understood, our universe can be accurately described by polynomial Hamiltonians of low order,” say Lin and Tegmark. Typically, the polynomials that describe laws of physics have orders ranging from 2 to 4.

The laws of physics have other important properties. For example, they are usually symmetrical when it comes to rotation and translation. Rotate a cat or dog through 360 degrees and it looks the same; translate it by 10 meters or 100 meters or a kilometer and it will look the same. That also simplifies the task of approximating the process of cat or dog recognition.

These properties mean that neural networks do not need to approximate an infinitude of possible mathematical functions but only a tiny subset of the simplest ones.

There is another property of the universe that neural networks exploit. This is the hierarchy of its structure. “Elementary particles form atoms which in turn form molecules, cells, organisms, planets, solar systems, galaxies, etc.,” say Lin and Tegmark. And complex structures are often formed through a sequence of simpler steps.

Now Lin and Tegmark say they’ve worked out why. The answer is that the universe is governed by a tiny subset of all possible functions. In other words, when the laws of physics are written down mathematically, they can all be described by functions that have a remarkable set of simple properties.

So deep neural networks don’t have to approximate any possible mathematical function, only a tiny subset of them.

To put this in perspective, consider the order of a polynomial function, which is the size of its highest exponent. So a quadratic equation like y=x2 has order 2, the equation y=x24 has order 24, and so on.

Obviously, the number of orders is infinite and yet only a tiny subset of polynomials appear in the laws of physics. “For reasons that are still not fully understood, our universe can be accurately described by polynomial Hamiltonians of low order,” say Lin and Tegmark. Typically, the polynomials that describe laws of physics have orders ranging from 2 to 4.

The laws of physics have other important properties. For example, they are usually symmetrical when it comes to rotation and translation. Rotate a cat or dog through 360 degrees and it looks the same; translate it by 10 meters or 100 meters or a kilometer and it will look the same. That also simplifies the task of approximating the process of cat or dog recognition.

These properties mean that neural networks do not need to approximate an infinitude of possible mathematical functions but only a tiny subset of the simplest ones.

There is another property of the universe that neural networks exploit. This is the hierarchy of its structure. “Elementary particles form atoms which in turn form molecules, cells, organisms, planets, solar systems, galaxies, etc.,” say Lin and Tegmark. And complex structures are often formed through a sequence of simpler steps.

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