We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the performance gap between pure machine learning approaches to that of the best numerical or hybrid solvers. This is achieved with new representations - separable spectral layers and improved residual connections - and a combination of training strategies such as the Markov assumption, Gaussian noise, and cosine learning rate decay. On several challenging benchmark PDEs on regular grids, structured meshes, and point clouds, the F-FNO can scale to deeper networks and outperform both the FNO and the geo-FNO, reducing the error by 83% on the Navier-Stokes problem, 31% on the elasticity problem, 57% on the airfoil flow problem, and 60% on the plastic forging problem. Compared to the state-of-the-art pseudo-spectral method, the F-FNO can take a step size that is an order of magnitude larger in time and achieve an order of magnitude speedup to produce the same solution quality.

翻译：我们提出因子化傅里叶神经算子（F-FNO），这是一种基于学习的偏微分方程（PDE）模拟方法。基于最近提出的流场傅里叶表示，F-FNO 弥合了纯机器学习方法与最优数值或混合求解器之间的性能差距。这一成果通过新的表示形式——可分离谱层与改进的残差连接——以及结合马尔可夫假设、高斯噪声和余弦学习率衰减等训练策略实现。在多个具有挑战性的基准PDE问题（涵盖规则网格、结构化网格和点云）上，F-FNO 可扩展至更深网络，并优于 FNO 和 geo-FNO，在Navier-Stokes问题中误差降低83%，弹性问题中降低31%，翼型流动问题中降低57%，塑性锻造问题中降低60%。与最先进的伪谱方法相比，F-FNO 在时间步长上可增大一个数量级，并在产生相同解质量的情况下实现一个数量级的加速。