MAMIKON GULIAN: I'm here to talk to you
about a new way of discovering physics
using contemporary methods from machine learning.
Of course, machine learning is an area of computer science
that's been recently developed to solve complex problems
in commercial industry, but we're
turning it around and using it to do fundamental science.
In the process we'll see how machine learning can actually
shed light on fractional calculus,
a powerful mathematical tool that was developed,
but then almost immediately forgotten over 300 years ago.
So to motivate you, all of the great technology
that we enjoy today is due to the discovery of physical laws.
These are mathematical models consisting
of rules and equations that govern our physical world.
I have a few of these great equations here.
So this is Newton's-- one of Newton's laws for classical
mechanics.
This is the heat or the diffusion equation
discovered by Fourier more than a century after Newton.
This is one of Maxwell's equations,
and these govern electromagnetism.
They are the basis for all the electrical engineering
and wireless technology that we have today.
And this is the Schroedinger equation, which
governs quantum mechanics.
So, for example, the particles in a nuclear reactor,
the electrons in your battery, the semiconductors that
make your computer possible.
Now, how are all these equations discovered
if they're so important?
Well, the process actually hasn't
changed that much since the days of Newton or even Archimedes.
So the problems are too complex to attack directly,
and mathematicians instead use their creativity
and their imagination to try to discover some key ideas that
will explain everything.
This creative process is a very risky guessing game.
It's trial and error.
It's a very indirect way to attack the problem.
So the researchers typically work in--
alone or in small groups, often in secrecy
to avoid ridicule or even theft of their ideas.
This is where machine learning comes in,
because machine learning is specifically
developed to attack complex problems directly,
and it's already being used today for utilitarian purposes.
For example, the movie recommendation system
on Netflix or online ads, and also
in much more critical applications
like medical and financial data.
But can it be used to discover physics as I suggest?
The answer is yes, and this is a very cutting edge
area at the intersection of math,
computer science, and physics.
The example that I have here is a work
by Michael Schmidt and Hod Lipson,
and they discovered the conservation law
of the double pendulum and other systems using machine learning.
The double pendulum is a textbook physical system.
What they did is they put sensors on it and let it spin.
It's a very chaotic system, it does all sorts of somersaults.
They measured the motion over time,
and then they crunch the data and discovered the conservation
law correctly.
And actually, Brown is a pioneer in this field.
And just to go back, so if you wanted to be a physicist,
you would train for years until you could actually
derive this conservation law that they
did from Newton's, as a special case of Newton's theory.
But they did it from scratch using only machine learning
with very few assumptions.
And as I said, Brown is a pioneer in this area.
It's being advanced by my friends Maziar, Paris,
and my advisor George under the heading of hidden physics
discovery from data.
And we use more of a Gaussian process approach.
What-- So this is the area that my research is in.
What I want to do specifically, is
to develop machine learning of fractional equations.
So what's that?
Well, what is fractional calculus?
Fractional calculus is calculus with fractional derivatives.
The idea is that from ordinary calculus,
you know that you have a first derivative, which
is like velocity and then a second derivative, which
is like acceleration.
It turns out you can have a derivative
of any order between one and two, in fact any order at all.
And the interesting thing is that these were discovered
by Leibniz in 1695, a few years after he
discovered ordinary calculus along with Newton.
But they were forgotten because the formulas
for these fractional derivatives are so complex, as you
can see behind me, it's almost impossible
to work with them using pencil and paper.
So no wonder they were forgotten then,
but now we have computers and machine learning,
and complexity is no longer a problem.
So-- And fractional calculus, in the past decade,
people have understood that fractional calculus
is the correct way to model anomalous phenomena.
A great example is anomalous diffusion,
which is a type of diffusion or you
can think like heat spreading, that
occurs in finance, in the plasma and fusion reactors,
in the flow of water underground.
It's everywhere, as evidenced by the title of this paper.
And machine learning is exactly the tool
we need to finally derive these models.
Now for the big picture, Eric Temple Bell, in his great book
Mathematics Queen And Servant of Science,
wrote that calculus is considered
to be the most powerful tool ever devised
for scientific thought.
But we hope to show in the long term
that it's just the tip of the iceberg,
and that fractional calculus, with some help from machine
learning, can lead to a new era in mathematical modeling.
Thank you.