论文标题
傅立叶神经操作员具有一般几何形状的PDE的学识板变形
Fourier Neural Operator with Learned Deformations for PDEs on General Geometries
论文作者
论文摘要
深度学习替代模型在解决偏微分方程（PDE）方面已显示出希望。其中，傅立叶神经操作员（FNO）达到了良好的准确性，并且与数值求解器（例如流体流量）上的数值求解器相比要快得多。但是，FNO使用快速傅立叶变换（FFT），该变换仅限于具有均匀网格的矩形域。在这项工作中，我们提出了一个新的框架，即Geo-Fno，以解决任意几何形状的PDE。 Geo-FNO学会将可能不规则的输入（物理）结构域变形为具有均匀网格的潜在空间。具有FFT的FNO模型在潜在空间中应用。由此产生的GEO-FNO模型既具有FFT的计算效率，也具有处理任意几何形状的灵活性。我们的Geo-fno也可以灵活地在其输入格式，即点云，网格和设计参数方面都是有效的输入。我们考虑了各种PDE，例如弹性，可塑性，Euler和Navier-Stokes方程，以及正向建模和逆设计问题。与标准数值求解器相比，Geo-fno的价格比标准数值求解器快$ 10^5 $倍，与在现有基于ML的PDE求解器（例如标准FNO）上进行直接插值相比，高度准确两倍。
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.
