Inverse Evolution Data Augmentation for Neural PDE Solvers

TL;DR

The paper addresses data scarcity in neural operators for evolution PDEs by introducing inverse-evolution–based data augmentation, which generates training pairs from the inverse process and enables high-order schemes that preserve implicit-discretization consistency. By swapping data pairs and using large time steps, the method yields efficient, diverse training data that reinforce the operator’s stability and accuracy. It also introduces preprocessing to handle sharp interfaces and analyzes spatial discretization choices to further stabilize augmentation. Empirical results on Burgers', Allen-Cahn, and Navier–Stokes show that coupling inverse-evolution augmentation with Fourier Neural Operator (and UNet) improves accuracy and robustness, particularly in data-limited settings, demonstrating practical impact for fast, reliable neural PDE solvers.

Abstract

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.

Figures (5)

• Figure 1: Comparison between inverse evolution and forward evolution on heat diffusion equation with $\Delta t =0.05$. Inverse evolution produces more accurate data pairs ($U_0, U_1$) as the order increases, while forward evolution yields inaccurate data pairs ($U_1, U_2$) across all schemes.
• Figure 2: Inverse evolution between original data and preprocessed data. The first row shows the inverse evolution of the original data, while the second row presents the results based on preprocessed data.
• Figure 3: Generated data for Allen-Cahn equation with $\epsilon=0.01$ and $\Delta t=0.5$. The first two columns are generated inputs and outputs from the inverse evolution, respectively. The last column shows the true solutions of the generated input.
• Figure 4: Generated data for Navier-Stokes equation with $\nu = 0.001$ and $\Delta t=0.5$. The first two columns are generated inputs and outputs from the inverse evolution, respectively. The last column shows the true solutions of the generated input.
• Figure 5: Predictions of FNO trained on different datasets for Allen-Cahn equation ($\epsilon = 0.05$). (a) inputs with noise (b) 1000 data pairs (c) 10000 data pairs (d) 1000 data pairs + 1000 generated data pairs (e) 5000 data pairs + 5000 generated data pairs (f) ground truth