Unless you are a physicist or a technology engineer, you don’t need to know what partial differential equations do. Even Karen Hao, MIT veteran AI correspondent, who was directly exposed to this equation when he was in engineering and is also the author original article on the MIT Technology Review, also never applied strange equations in real life.
However, the partial differential equation, or PDE for short, still exists and is still a very little man-made magic. PDE is a mathematical equation that effectively describes changes taking place in space and time, so humans still use PDE to describe physical phenomena in the Universe. We can apply PDE to a variety of models, from the way the planets orbit, the displacement of geological arrays to the turbulence of air around the body of an aircraft in the air, to predict events will take place while designing the aircraft to be safe.
PDE is very difficult to solve, let me give an example to explain the concept of “solving” in this case. Imagine you are modeling the turbulence of the air to try out a new model plane. We have a PDE equation called Navier-Stokes to describe the behavior of any solution. “Solving” the Naniver-Stokes equation, you get a pattern of the motion of the air at a given moment, and based on that model of how the air current will move in the future, or how the wind was blowing in the previous moment.
Former US President Barrack Obama holds a shirt that says both the Navier-Stokes equation and the Moses story: “Then Moses said: ‘Navier-Stokes equation’. And the country split into two. ! “.
These calculations are complex and computationally resource intensive, which is why we use a supercomputer to model rules that require many PDE equations. This is also the reason why the AI industry is interested in studying these equations. If we can use deep learning to speed up PDE solving, we can accelerate the speed of science and engineering development.
Recently, researchers at Caltech announced a new deep learning technique that allows computers to solve PDE much more accurately than ever before. The new modal nature is also more general, allowing us to solve a whole set of PDE equations – for example the Navier-Stokes equation applied to any type of solution – without having to retrain the AI with a set other data. Finally, the speed of solving PDE is 1,000 times faster than in the past, helping researchers to no longer depend on supercomputers, while also helping us to solve more complex problems.
Seemed irrelevant: The legendary rapper congratulated
Before you learn how the researchers solved PDE, be overwhelmed with the results the team achieved. In the gif below, you can see a description of what the team of scientists achieved. The first column shows the movement of the solution at a given point; the second column shows the actual motion solution, translating with the flow of time; the third column is the neural net’s prediction of the possible motion of the solution.
You can see the obvious similarities of column 2 and column 3; The artificial intelligence system has successfully predicted the way the solution moves.
Twitter users have been elated by the new research report, even by legendary rapper and producer MC Hammer Also posted to congratulate the team of scientists.
Here’s how they do it
The first thing to understand: basically, a neural net is a basic function estimator. When training the system with a data set of input and output pairs, researchers actually try to compute a function – a series of calculations – that can turn input data into output. .
For example, we have a cat identification machine in our hands. You will train the neural net by feeding into the system of cat images and unrelated images – all of these data are “inputs”, and mark them with the letters 1 and 0 – this is ” output”. The neural net will find the most plausible function that can turn each input image into the value “1”, and all unrelated data to “0”. By the main function the system finds, it will recognize the cat’s subject as it continues to look at new images. If the training goes smoothly, the system will have a high rate of accurate cat recognition.
The process of finding this function is an important factor for researchers to solve the PDE equation. The ultimate goal is to find a function that most accurately describes the motion of air particles in space and time.
This is the key point raised by the scientific report. Typically, neural networks are used to approximate functions that convert input data into outputs in Euclidean space – a common graph with x, y and z axes. But this time, the scientists decided to determine the input and output values in Fourier space, which is a commonly used graph in describing wave frequencies. With support from other studies in the other array, which suggested that air motion can be described as a collection of a series of wave frequencies, the scientists decided to separate from the old thinking. The usual macroscopic wind movement will be similar to long and low frequency waves, while occasional microscopic winds will be high frequency waves, short and fast.
The illustration of air disturbance.
The key point of these factors? Because in Fourier space, it is easier to estimate the Fourier function than solving the PDE in Euclidean space, thereby simplifying the task of neural networks to perform. Besides the faster calculation speed, the new method of solving the Navier-Stokes equation yields about 30% fewer errors than previous deep learning methods.
This solution is brilliantly ingenious, but it can be applied more widely. Previous deep learning methods required researchers to use one set of data for each different solution, and the new system only needed one training to find the motion of all solutions. Although the team hasn’t worked on many other things yet, the current capabilities would allow the system to solve PDE equations related to seismic activity or the ability to transfer heat in matter.
A super emulator system
Head of Research, Professor Anima Anandkumar.
The team doesn’t write reports for fun, they want to use AI in more scientific aspects. Thanks to the cooperation from climate science, seismic science, and material science, head of research team Anima Anandkumar and colleagues dare to try to solve the PDE equation. Currently, they are working on a new neural network application with experts from Caltech and Lawrence Berkeley National Laboratory.
Professor Anima Anandkumar is very interested in climate change research. The Navier-Stokes equation not only predicts air turbulence well, it also models weather fluctuations.
“It is difficult to have good weather forecasting on a global scale is still a great difficulty. Even using a supercomputer, it’s impossible to make predictions on such a large scale. If we could apply this method to speed up the whole process, the impact would be huge“, Ms. Anandkumar said, emphasizing that these are not the only applications of the new way to solve the PDE function.
Refer to MIT Technology Review