用逆演化方法高效生成训练数据，提升神经PDE求解器性能

Neural networks have emerged as promising tools for solving partial differential equations (PDEs), particularly through the application of neural operators. Training neural operators typically requires a large amount of training data to ensure accuracy and generalization. In this paper, we propose a novel data augmentation method specifically designed for training neural operators on evolution equations. Our approach utilizes insights from inverse processes of these equations to efficiently generate data from random initialization that are combined with original data. To further enhance the accuracy of the augmented data, we introduce high-order inverse evolution schemes. These schemes consist of only a few explicit computation steps, yet the resulting data pairs can be proven to satisfy the corresponding implicit numerical schemes. In contrast to traditional PDE solvers that require small time steps or implicit schemes to guarantee accuracy, our data augmentation method employs explicit schemes with relatively large time steps, thereby significantly reducing computational costs. Accuracy and efficacy experiments confirm the effectiveness of our approach. Additionally, we validate our approach through experiments with the Fourier Neural Operator and UNet on three common evolution equations that are Burgers' equation, the Allen-Cahn equation and the Navier-Stokes equation. The results demonstrate a significant improvement in the performance and robustness of the Fourier Neural Operator when coupled with our inverse evolution data augmentation method.