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.

本论文提出了基于深度学习的 Fourier 神经算子 (FNO) 的新框架 - geo-FNO, 用于求解偏微分方程，并可在任意几何图形上工作。该方法利用深度学习降噪算法，将不规则的物理空间映射到一个均匀的潜空间中，再应用 FNO 模型来求解偏微分方程。geo-FNO 比传统的数值求解方法快 $10^5$ 倍，比其它机器学习算法（如 FNO）的直接插值更准确。