Fourier transforms reveal how AI learns advanced physics

Scientific AI’s black field isn’t any match for the 200-year-old methodology

Fourier transforms reveal how the deep neural community learns advanced physics.

One of many oldest instruments in computational physics, a 200-year-old mathematical approach often known as Fourier evaluation might reveal essential insights into how a type of synthetic intelligence known as a deep neural community learns to carry out duties involving advanced physics akin to modeling climate and turbulence, in line with a brand new examine.

The invention by mechanical engineering researchers at Rice College is described in an open entry examine revealed within the journal PNAS nexusa sister publication of Proceedings of the Nationwide Academy of Sciences.

That is the primary rigorous framework to clarify and information using deep neural networks for advanced dynamical methods akin to local weather, mentioned examine correspondent creator Pedram Hassanzadeh. It may considerably speed up using scientific deep studying in local weather science and result in way more dependable local weather change projections.

Researchers at Rice College have skilled a type of synthetic intelligence known as a deep studying neural community to acknowledge advanced flows of air or water and predict how the flows will change over time. This visible illustrates the dramatic variations within the scale of options proven to the mannequin throughout coaching (high) and the options it learns to acknowledge (backside) to make predictions. Credit score: Picture courtesy of P. Hassanzadeh/Rice College

Within the paper, Hassanzadeh, Adam Subel and Ashesh Chattopadhyay, each alumni, and Yifei Guan, a postdoctoral analysis affiliate, detailed their use of Fourier evaluation to check a deep studying neural community that was skilled to acknowledge advanced flows of air within the ambiance or water within the ocean and predict how these flows will change over time. Their evaluation revealed not solely what the neural community had realized, but in addition allowed us to instantly relate what the community had realized to the physics of the advanced system it was modeling, Hassanzadeh mentioned.

Deep neural networks are notoriously obscure and are sometimes considered black bins, he mentioned. It is a main concern with utilizing deep neural networks in scientific functions. The opposite is generalizability: these networks can not work for a system aside from the one they have been skilled for.

Coaching state-of-the-art deep neural networks requires a considerable amount of knowledge, and the retraining burden with present strategies continues to be important. After coaching and retraining a deep studying community to carry out numerous duties involving advanced physics, researchers at Rice College used Fourier evaluation to check all 40,000 kernels from the 2 iterations and located that greater than 99% have been comparable. This illustration exhibits the Fourier spectra of the 4 kernels that differed essentially the most earlier than (left) and after (proper) retraining. The outcomes exhibit the potential of strategies to establish extra environment friendly routes to retraining that require considerably much less knowledge. Credit score: Picture courtesy of P. Hassanzadeh/Rice College

Hassanzadeh mentioned the analytical framework introduced by his crew within the paper opens up the black field, permits us to look inside to grasp what the networks realized and why, and likewise permits us to narrate it to the physics of the system that was realized.

Subel, the examine’s lead creator, started the analysis as a Rice undergraduate and is now a graduate pupil at

” data-gt-translate-attributes=”[{” attribute=””>New York University. He said the framework could be used in combination with techniques for transfer learning to enable generalization and ultimately increase the trustworthiness of scientific deep learning.

While many prior studies had attempted to reveal how deep learning networks learn to make predictions, Hassanzadeh said he, Subel, Guan and Chattopadhyay chose to approach the problem from a different perspective.

Pedram Hassanzadeh. Credit: Rice Universit

The common

He said Fourier analysis, which was first proposed in the 1820s, is a favorite technique of physicists and mathematicians for identifying frequency patterns in space and time.

People who do physics almost always look at data in the Fourier space, he said. It makes physics and math easier.

For example, if someone had a minute-by-minute record of outdoor temperature readings for a one-year period, the information would be a string of 525,600 numbers, a type of data set physicists call a time series. To analyze the time series in Fourier space, a researcher would use trigonometry to transform each number in the series, creating another set of 525,600 numbers that would contain information from the original set but look quite different.

Instead of seeing temperature at every minute, you would see just a few spikes, Subel said. One would be the cosine of 24 hours, which would be the day and night cycle of highs and lows. That signal was there all along in the time series, but Fourier analysis allows you to easily see those types of signals in both time and space.

Based on this method, scientists have developed other tools for time-frequency analysis. For example, low-pass transformations filter out background noise, and high-pass filters do the inverse, allowing one to focus on the background.

Adam Subel. Credit: Rice University

Hassanzadehs team first performed the Fourier transformation on the equation of its fully trained deep-learning model. Each of the models approximately 1 million parameters act like multipliers, applying more or less weight to specific operations in the equation during model calculations. In an untrained model, parameters have random values. These are adjusted and honed during training as the algorithm gradually learns to arrive at predictions that are closer and closer to the known outcomes in training cases. Structurally, the model parameters are grouped in some 40,000 five-by-five matrices, or kernels.

When we took the Fourier transform of the equation, that told us we should look at the Fourier transform of these matrices, Hassanzadeh said. We didnt know that. Nobody has done this part ever before, looked at the Fourier transforms of these matrices and tried to connect them to the physics.

And when we did that, it popped out that what the neural network is learning is a combination of low-pass filters, high-pass filters and Gabor filters, he said.

The beautiful thing about this is, the neural network is not doing any magic, Hassanzadeh said. Its not doing anything crazy. Its actually doing what a physicist or mathematician might have tried to do. Of course, without the power of neural nets, we did not know how to correctly combine these filters. But when we talk to physicists about this work, they love it. Because they are, like, Oh! I know what these things are. This is what the neural network has learned. I see.

Subel said the findings have important implications for scientific deep learning, and even suggest that some things scientists have learned from studying machine learning in other contexts, like classification of static images, may not apply to scientific machine learning.

We found that some of the knowledge and conclusions in the machine learning literature that were obtained from work on commercial and medical applications, for example, do not apply to many critical applications in science and engineering, such as climate change modeling, Subel said. This, on its own, is a major implication.

Reference: Explaining the physics of transfer learning in data-driven turbulence modeling by Adam Subel, Yifei Guan, Ashesh Chattopadhyay and Pedram Hassanzadeh, 23 January 2023, PNAS Nexus.
DOI: 10.1093/pnasnexus/pgad015

Chattopadhyay received his Ph.D. in 2022 and is now a research scientist at the Palo Alto Research Center.

The research was supported by the Office of Naval Research (N00014- 20-1-2722), the National Science Foundation (2005123, 1748958) and the Schmidt Futures program. Computational resources were provided by the National Science Foundation (170020) and the National Center for Atmospheric Research (URIC0004).