“Fourier Neural Operator for Parametric Partial Differential Equations,” by Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar, was posted to arXiv on October 18, 2020, and presented at ICLR 2021. It introduced the Fourier Neural Operator, or FNO, a network that learns a mapping between entire functions rather than between fixed-size vectors.
The key idea is to do the heavy lifting in Fourier space. Each layer transforms its input into the frequency domain, multiplies by a set of learned weights on the lowest frequencies, and transforms back. Because this operation is defined on functions and not on a particular grid, the same trained model can be evaluated at any resolution, including resolutions finer than anything it was trained on, a property the authors call zero-shot super-resolution.
The practical payoff is speed. The paper reported that for turbulent flows governed by the Navier-Stokes equations the FNO produced solutions up to three orders of magnitude faster than traditional numerical solvers, while learning a whole parametric family of equations at once instead of solving a single instance. The same architecture later became the backbone of fast weather and climate emulators.
For a general reader, the FNO is a clean example of the surrogate-model idea: instead of running a slow, exact simulation every time, you train a fast approximate one that learns the underlying operator. When the simulation is the bottleneck, as it is in weather, fluids, and engineering design, that speedup changes what is practical to compute.