Fourier Neural Operator for Parametric Partial Differential Equations

The Fourier Neural Operator (FNO) extends neural networks beyond mappings between finite-dimensional Euclidean spaces to operators that map between function spaces. For partial differential equations, it directly learns the mapping from functional parametric inputs to solutions, so a single trained model captures an entire family of PDEs rather than one instance. The core idea is to parameterize the integral kernel directly in Fourier space, producing an expressive yet computationally efficient architecture.

The authors evaluate FNO on Burgers' equation, Darcy flow, and the Navier-Stokes equation, including the turbulent regime. It achieves state-of-the-art performance relative to existing neural network methods and runs up to three orders of magnitude faster than traditional numerical PDE solvers. This combination of accuracy and speed positioned it as a fast, accurate surrogate for simulating physical systems governed by PDEs.