Fourier Neural Operator for Parametric Partial Differential Equations

The classical development of neural networks has primarily focused on
learning mappings between finite-dimensional Euclidean spaces. Recently, this
has been generalized to neural operators that learn mappings between function
spaces. For partial differential equations (PDEs), neural operators directly
learn the mapping from any functional parametric dependence to the solution.
Thus, they learn an entire family of PDEs, in contrast to classical methods
which solve one instance of the equation. 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.

Authors

Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar