Reachability and asymptotics of Gaussian Transformer dynamics

Researchers have formulated Transformer data propagation as a nonlinear control system and proven that Gaussian distributions remain Gaussian through the network's layers. This reduces infinite-dimensional dynamics to finite-dimensional equations governing mean and covariance evolution, connecting Transformer expressiveness to classical control theory and revealing conditions for stability or divergence.

This theoretical paper advances understanding of how Transformers—the architecture underlying large language models—process information through mathematical rigor rather than empirical observation. By modeling data flow as a control system on probability measures, the researchers prove that Gaussian input distributions maintain their Gaussian form across layers, a property that enables dramatic simplification of the system's dynamics. This invariance property transforms what would otherwise be an intractable infinite-dimensional problem into a manageable finite-dimensional bilinear control system, reducing computational complexity while preserving theoretical accuracy.

The work establishes explicit spectral conditions determining whether a Transformer's covariance matrix evolves toward stable equilibria or diverges catastrophically—conditions grounded in classical Riccati equations from filtering and control theory. This connection illuminates why certain architectural configurations produce bounded, interpretable outputs while others fail. The reachability framework reformulates core questions about Transformer expressiveness: can the network reach prescribed Gaussian moment targets? The authors prove exact finite-time reachability is possible for target distributions whose covariance rank matches the initial distribution, revealing an intrinsic invariant constraint.

Numerical validation demonstrates that practical Transformers with Gaussian inputs remain close to theoretical moment-matched distributions through early and intermediate layers, lending empirical credibility to the theoretical predictions. The identification of destabilizing configurations that produce covariance blow-up has potential implications for understanding failure modes in large language models. For the AI research community, this mathematical framework provides tools to analyze Transformer behavior rigorously, potentially informing better architectural design and training procedures. The bridge to classical control theory opens pathways for applying decades of control-theoretic insights to modern deep learning architecture design.

• →Gaussian distributions provably remain Gaussian through Transformer layers, reducing infinite-dimensional dynamics to finite-dimensional covariance evolution equations.
• →The framework reformulates Transformer expressiveness as a reachability problem connected to classical Riccati equations from control theory and filtering.
• →Explicit spectral conditions determine whether Transformer covariance matrices achieve stable equilibria or exhibit finite-time blow-up behavior.
• →Numerical experiments confirm theoretical predictions for practical Transformers with Gaussian inputs across early and intermediate layers.
• →The rank invariant of initial covariance fundamentally constrains which target Gaussian distributions a Transformer can reach.