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.

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Spiny Norman said:

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.

More

https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations

Ta, something didn’t copy fully when I pasted the link.

dv said:

Spiny Norman said:

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.

More

https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations

Will read.

Navier-Stokes equation was my bread and butter for many years.

As someone succinctly put it at a conference I attended.
“If it were any harder it’d be impossible, and if it were any easier then we wouldn’t have a job.”

Some other partial differential equations come in very useful, too.

> Now researchers at Caltech have introduced a new deep-learning technique for solving PDEs that is dramatically more accurate than deep-learning methods developed previously. It’s also much more generalizable, capable of solving entire families of PDEs—such as the Navier-Stokes equation for any type of fluid—without needing retraining. Finally, it is 1,000 times faster than traditional mathematical formulas, which would ease our reliance on supercomputers and increase our computational capacity to model even bigger problems. That’s right. Bring it on.

Yike. Wow. I approve.

> Now here’s the crux of the paper. Neural networks are usually trained to approximate functions between inputs and outputs defined in Euclidean space, your classic graph with x, y, and z axes. But this time, the researchers decided to define the inputs and outputs in Fourier space, which is a special type of graph for plotting wave frequencies.

That could work. For simple geometries.

> One research topic Anandkumar is particularly excited about: climate change. Having good, fine-grained weather predictions on a global scale is such a challenging problem. So if we can use these methods to speed up the entire pipeline, that would be tremendously impactful.

Earth’s atmosphere is a simple geometry. I don’t see any reason why that wouldn’t be as accurate as the much slower methods in use for global climate modelling today.

Many many years ago, like in the 1970s and 1980s, there was a lot of effort put into solving Navier-Stokes equation in Fourier coordinates rather than the Cartesian coordinates that I used and are so popular today. The main reason for using Cartesian coordinates is the relative ease of handling geometrically complicated boundaries.

mollwollfumble said:

> Now researchers at Caltech have introduced a new deep-learning technique for solving PDEs that is dramatically more accurate than deep-learning methods developed previously. It’s also much more generalizable, capable of solving entire families of PDEs—such as the Navier-Stokes equation for any type of fluid—without needing retraining. Finally, it is 1,000 times faster than traditional mathematical formulas, which would ease our reliance on supercomputers and increase our computational capacity to model even bigger problems. That’s right. Bring it on.

Yike. Wow. I approve.

> Now here’s the crux of the paper. Neural networks are usually trained to approximate functions between inputs and outputs defined in Euclidean space, your classic graph with x, y, and z axes. But this time, the researchers decided to define the inputs and outputs in Fourier space, which is a special type of graph for plotting wave frequencies.

That could work. For simple geometries.

> One research topic Anandkumar is particularly excited about: climate change. Having good, fine-grained weather predictions on a global scale is such a challenging problem. So if we can use these methods to speed up the entire pipeline, that would be tremendously impactful.

Earth’s atmosphere is a simple geometry. I don’t see any reason why that wouldn’t be as accurate as the much slower methods in use for global climate modelling today.

Many many years ago, like in the 1970s and 1980s, there was a lot of effort put into solving Navier-Stokes equation in Fourier coordinates rather than the Cartesian coordinates that I used and are so popular today. The main reason for using Cartesian coordinates is the relative ease of handling geometrically complicated boundaries.

Well I remain somewhat sceptical, but we shall see.

It’s about time there was something new in this area.