This paper proposes a novel approach to integrating partial differential equation (PDE)-based evolution models into neural networks through a new type of regularization. Specifically, we propose inverse evolution layers (IELs) based on evolution equations. These layers can achieve specific regularization objectives and endow neural networks' outputs with corresponding properties of the evolution models. Moreover, IELs are straightforward to construct and implement, and can be easily designed for various physical evolutions and neural networks. Additionally, the design process for these layers can provide neural networks with intuitive and mathematical interpretability, thus enhancing the transparency and explainability of the approach. To demonstrate the effectiveness, efficiency, and simplicity of our approach, we present an example of endowing semantic segmentation models with the smoothness property based on the heat diffusion model. To achieve this goal, we design heat-diffusion IELs and apply them to address the challenge of semantic segmentation with noisy labels. The experimental results demonstrate that the heat-diffusion IELs can effectively mitigate the overfitting problem caused by noisy labels.

翻译：本文提出一种通过新型正则化将基于偏微分方程（PDE）的演化模型集成到神经网络中的创新方法。具体而言，我们基于演化方程提出了逆演化层（IELs）。这些层能够实现特定的正则化目标，并使神经网络的输出具备相应演化模型的特性。此外，IELs构造与实现过程简洁直接，易于针对多种物理演化过程和神经网络进行设计。更重要的是，这些层的设计流程可为神经网络提供直观的数学可解释性，从而提升方法的透明度和可解释性。为验证本方法的有效性、高效性与简易性，我们以热扩散模型为例，赋予语义分割模型平滑性特征。为此，我们设计了热扩散逆演化层，并将其应用于含噪声标签的语义分割挑战。实验结果表明，热扩散IELs能有效缓解噪声标签导致的过拟合问题。