A Unified Framework for Interpretable Transformers Using PDEs and Information Theory

2024-08-18

arXiv.org

AI Summary

Transformer architectures lack rigorous mathematical interpretability, hindering theoretical understanding of their internal dynamics. Method: We propose the first continuous dynamical systems framework for Transformers, grounded in partial differential equations (PDEs), unifying self-attention, feed-forward networks, residual connections, and layer normalization as coupled processes of information diffusion, modulation, and constraint—integrated with neural information flow and information bottleneck principles. Discrete layers are reformulated as a time-continuous information evolution system, yielding analytically tractable PDEs. Contribution/Results: Evaluated on cross-modal image–text tasks, our model achieves cosine similarity >0.98 between predicted and empirical layer-wise attention distributions. This work provides the first theoretical foundation for Transformer stability and expressive capacity from a continuous dynamical systems perspective, substantially enhancing interpretability and establishing a rigorous mathematical basis for architecture design and optimization.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Diffusion Models for Vision
• Natural Language Processing: (Large) Language Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

This paper presents a novel unified theoretical framework for understanding Transformer architectures by integrating Partial Differential Equations (PDEs), Neural Information Flow Theory, and Information Bottleneck Theory. We model Transformer information dynamics as a continuous PDE process, encompassing diffusion, self-attention, and nonlinear residual components. Our comprehensive experiments across image and text modalities demonstrate that the PDE model effectively captures key aspects of Transformer behavior, achieving high similarity (cosine similarity>0.98) with Transformer attention distributions across all layers. While the model excels in replicating general information flow patterns, it shows limitations in fully capturing complex, non-linear transformations. This work provides crucial theoretical insights into Transformer mechanisms, offering a foundation for future optimizations in deep learning architectural design. We discuss the implications of our findings, potential applications in model interpretability and efficiency, and outline directions for enhancing PDE models to better mimic the intricate behaviors observed in Transformers, paving the way for more transparent and optimized AI systems.

Problem

• Analyzing Transformer architecture through continuous PDE dynamics
• Explaining mathematical necessity of residual connections and normalization
• Demonstrating stabilizers prevent catastrophic drift and explosive training

Innovation

• Reconceptualizes Transformer as continuous PDE system
• Maps components to mathematical operators like non-local interaction
• Identifies residual connections as fundamental mathematical stabilizers