Reachability and asymptotics of Gaussian Transformer dynamics

2026-05-29 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, quick

Summary

The paper formulates data propagation within the Transformer architecture, which powers large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, it proves that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing mean and covariance evolution, reframing Transformer expressive capacity as a reachability problem for Gaussian moments, and connecting it to Riccati-type equations. For time-varying controls, exact finite-time reachability of any target Gaussian distribution is proven, provided its covariance matrix has the same rank as the initial one. Time-invariant parameters yield explicit spectral conditions for asymptotic stability or finite-time covariance blow-up. Numerical experiments confirm that practical Transformers with Gaussian inputs stay close to moment-matched Gaussian distributions in early and intermediate layers.

Key takeaway

For AI Scientists analyzing or designing Transformer architectures, understanding the dynamics of data propagation is crucial. This work reveals that Gaussian inputs maintain their Gaussian nature, simplifying the complex infinite-dimensional dynamics to a finite-dimensional bilinear control system. You should leverage this framework to predict the stability of your models and assess the reachability of desired data distributions, especially when dealing with Gaussian-like data.

Key insights

Gaussian distributions remain invariant through mean-field Transformer dynamics, simplifying their analysis to finite-dimensional systems.

Principles

• Gaussian invariance reduces infinite-dimensional dynamics to finite-dimensional systems.
• Covariance matrix rank is an intrinsic invariant of Transformer dynamics.
• Spectral conditions determine asymptotic stability or finite-time covariance blow-up.

Method

Formulating Transformer data propagation as a nonlinear control system on probability measures, then reducing it to a finite-dimensional bilinear control system for Gaussian moments.

In practice

• Analyze Transformer expressive capacity via Gaussian reachability problems.
• Predict covariance evolution and stability regimes in Transformer models.
• Inform design of stable Transformer configurations based on spectral conditions.

Topics

• Transformer Dynamics
• Gaussian Distributions
• Mean-field Theory
• Control Systems
• Riccati Equations
• Covariance Evolution
• Reachability Analysis

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.