Researchers at the California Institute of Technology have introduced a new deep learning architecture – Fourier neural operator – for solving parametric partial differential equations (PDEs).

This new technique offers tremendous speed and accuracy advantage which, when coupled with its ability to solve entire families of PDEs, can boost our computational capacity to model crucial problems related to physical phenomena. Some practical applications can be in the areas of seismology, air turbulence and climate change – to predict seismic activity, design safer planes, and predict weather patterns on a global scale.

PDEs are extremely complicated equations that necessitate the use of supercomputers to solve the complex mathematical calculations involved. Notably, the function approximation between inputs and outputs by neural networks finds useful application in solving PDEs so this research seeks to bring AI to more scientific disciplines. The Fourier neural operator is capable of learning mappings between infinite-dimensional spaces of functions and fares far superior than all existing DL methods of solving these tricky equations.

 “In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers’ equation, Darcy flow, and the Navier-Stokes equation (including the turbulent regime). Our Fourier neural operator shows state-of-the-art performance compared to existing neural network methodologies and it is up to three orders of magnitude faster compared to traditional PDE solvers,” the paper states.

The research has been spearheaded by Caltech professors Anima Anandkumar, Andrew Stuart and Kaushik Bhattacharya. Their paper states: “Machine learning methods hold the key to revolutionizing many scientific disciplines by providing fast solvers that approximate traditional ones.”

Going further, this new approach of operator learning can also be extended beyond PDEs to cover problems in fields such as computer vision.