Flowing Through Layers: A Continuous Dynamical Systems Perspective on Transformers

TL;DR

This paper interprets transformer layers as a discretization of a continuous dynamical system, providing theoretical insights into their stability, convergence, and potential for architectural improvements.

Contribution

It introduces a continuous dynamical systems perspective on transformers, proving convergence and stability under certain conditions, and linking their behavior to ODEs.

Findings

Token representations converge to an ODE solution as layers increase.

Under one-sided Lipschitz conditions, dynamics are contractive and perturbations decay exponentially.

Provides a theoretical foundation connecting transformers to dynamical systems theory.

Abstract

We show that the standard discrete update rule of transformer layers can be naturally interpreted as a forward Euler discretization of a continuous dynamical system. Our Transformer Flow Approximation Theorem demonstrates that, under standard Lipschitz continuity assumptions, token representations converge uniformly to the unique solution of an ODE as the number of layers grows. Moreover, if the underlying mapping satisfies a one-sided Lipschitz condition with a negative constant, the resulting dynamics are contractive, causing perturbations to decay exponentially across layers. Beyond clarifying the empirical stability and expressivity of transformer models, these insights link transformer updates to a broader iterative reasoning framework, suggesting new avenues for accelerated convergence and architectural innovations inspired by dynamical systems theory.