The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as data augmentation, adversarial training, and noise injection, may encounter limited generalization due to model non-smoothness. In this paper, we propose to investigate generalization from a Partial Differential Equation (PDE) perspective, aiming to enhance it directly through the underlying function of neural networks, rather than focusing on adjusting input data. Specifically, we first establish the connection between neural network generalization and the smoothness of the solution to a specific PDE, namely ``transport equation''. Building upon this, we propose a general framework that introduces adaptive distributional diffusion into transport equation to enhance the smoothness of its solution, thereby improving generalization. In the context of neural networks, we put this theoretical framework into practice as PDE+ (\textbf{PDE} with \textbf{A}daptive \textbf{D}istributional \textbf{D}iffusion) which diffuses each sample into a distribution covering semantically similar inputs. This enables better coverage of potentially unobserved distributions in training, thus improving generalization beyond merely data-driven methods. The effectiveness of PDE+ is validated in extensive settings, including clean samples and various corruptions, demonstrating its superior performance compared to SOTA methods.

该论文提出了一种利用偏微分方程来改进神经网络泛化性能的方法，并将其实现为PDE + (具有自适应分布扩散的PDE)，该方法通过扩散每个样本以覆盖语义相似的输入分布，以改善泛化性能。

PDE+：通过自适应分布扩散 PDE 增强泛化能力