Flowing Through Layers: A Continuous Dynamical Systems Perspective on Transformers
This provides theoretical insights into transformer stability and expressivity, potentially enabling new architectural designs for machine learning researchers.
The paper shows that transformer layers can be interpreted as discretizations of continuous dynamical systems, proving that token representations converge to an ODE solution as layers increase and perturbations decay exponentially under certain conditions.
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.