PDE-Based (Partial Differential Equation) Models for Understanding Information Flow in Deep Neural Networks

Abstract

Transplanting the deep neural networks into high-stakes decision-making has left folks more and more willing to have a clear mathematically-founded understanding of how they actually operate. Researchers have been advocating the use of models that are not only transparent but also mathematically sound. The existing post-hoc explanation methods, saliency maps, LIME, SHAP, are all quite local. They are just veneer deceptions that leap out of a given layer and are not a degradation of actually unpacking the dynamics of feature evolution in the entire architecture. The framework that I would be suggesting here is based on Partial Differential Equations (PDEs). It is designed to model the information dynamics of neural nets, based on transport, diffusion, and non-local interaction equations. With the depth of a network as a sort of time axis, and channel activations as spatial fields, we can study an entire space of popular architectures: CNNs, ResNets, and transformers in a single PDE story. That allows one to discuss the concepts of stability, smoothness, propagation and a host of other properties using diffusion coefficients, velocity fields and non-local kernels that are mathematically characterised. We have tried the concept a little with a bit of theory and some cases, and it can warn you about structural bottlenecks, instabilities, over-smoothing, and representation collapse, the problems which you cannot pick at all when you are just looking at discrete layers. Concisely, interpreted via PDEs, PDE based interpretability provides a single, model free, mathematically sound view that enhances transparency, robustness, and design decisions. This article makes PDE modeling an indispensable method of research and interpretation of modern AI systems, and has consequences in future studies of hybrid ODEPDE models, physics-inspired architectures, and responsible AI.