Fourier Neural Operator for Parametric Partial Differential Equations

Inside the paper

The problem

Many scientific tasks require solving a family of partial differential equations for different inputs. A conventional numerical solve is repeated for each new instance, which can become costly when many solutions are needed.

The idea

The authors learn an operator mapping input functions to solution functions and implement its integral kernel in Fourier space. The learned mapping is applied across different discretizations in their experiments.

How it works

• Training pairs provide examples of input fields and corresponding solutions for a family of equations.
• The model lifts the input into a higher-dimensional representation and repeatedly applies layers that mix information across spatial positions.
• Each Fourier layer transforms features into frequency space, applies learned weights to retained modes, and transforms the result back before a nonlinearity.
• A final projection produces the predicted solution field, and the same learned parameters can be evaluated on a finer grid.

What it showed

The paper compares the method with other learned and numerical approaches on Burgers, Darcy flow and Navier–Stokes problems. It also reports a super-resolution experiment in which a model trained at one resolution is evaluated on a finer one.

Read it for

Read Section 2 for the operator-learning setup and Section 4 for the Fourier layer. Section 5.3 and Section 5.4 show the fluid and resolution experiments; the appendix contains more comparisons and implementation details that can wait.

Written in our own words from the paper itself. A draft the editors have not yet reviewed.