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.

翻译：深度学习代理模型在求解偏微分方程方面展现出潜力。其中，傅里叶神经算子（FNO）在多种偏微分方程（如流体流动）上实现了良好的精度，且相比数值求解器显著加快。然而，FNO使用快速傅里叶变换（FFT），这仅限于具有均匀网格的矩形域。在本工作中，我们提出了一种新框架，即geo-FNO，用于求解任意几何形状上的偏微分方程。Geo-FNO学习将输入（物理）域（可能是不规则的）形变为具有均匀网格的潜在空间。在潜在空间中应用基于FFT的FNO模型。由此产生的geo-FNO模型兼具FFT的计算效率和处理任意几何形状的灵活性。我们的geo-FNO在输入格式方面也具有灵活性，即点云、网格和设计参数均有效输入。我们考虑了多种偏微分方程，如弹性力学、塑性力学、欧拉方程和纳维-斯托克斯方程，以及正演建模和逆设计问题。Geo-FNO比标准数值求解器快10^5倍，并且相比于现有基于机器学习的偏微分方程求解器（如标准FNO）的直接插值方法，精度提升两倍。