Convection-Diffusion Equation: A Theoretically Certified Framework for Neural Networks

Differential equations have demonstrated intrinsic connections to network structures, linking discrete network layers through continuous equations. Most existing approaches focus on the interaction between ordinary differential equations (ODEs) and feature transformations, primarily working on input signals. In this paper, we study the partial differential equation (PDE) model of neural networks, viewing the neural network as a functional operating on a base model provided by the last layer of the classifier. Inspired by scale-space theory, we theoretically prove that this mapping can be formulated by a convection-diffusion equation, under interpretable and intuitive assumptions from both neural network and PDE perspectives. This theoretically certified framework covers various existing network structures and training techniques, offering a mathematical foundation and new insights into neural networks. Moreover, based on the convection-diffusion equation model, we design a new network structure that incorporates a diffusion mechanism into the network architecture from a PDE perspective. Extensive experiments on benchmark datasets and real-world applications confirm the effectiveness of the proposed model.