Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

TL;DR

This paper introduces geo-FNO, a neural network framework that efficiently solves PDEs on arbitrary geometries by learning domain deformations, combining the speed of Fourier neural operators with geometric flexibility.

Contribution

The paper proposes geo-FNO, a novel method that deforms irregular domains into a uniform grid for Fourier neural operator application, enabling fast and flexible PDE solving on complex geometries.

Findings

Geo-FNO is 100,000 times faster than traditional numerical solvers.

It is twice as accurate as existing ML-based PDE solvers like standard FNO.

Handles various PDEs and input formats such as point clouds and meshes.

Abstract

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.