  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.
