Continuous Dynamical Systems in Transformers

• The paper models Transformer layers as discretized ODE systems using Lie-Trotter and Strang-Marchuk splitting, clarifying numerical integration and convergence.
• It views Transformer operations as mean-field particle systems governed by Vlasov-type transport PDEs, explaining clustering and consensus among token representations.
• The work leverages operator-theoretic and variational principles to interpret layer normalization and energy minimization, guiding robust and interpretable model designs.

Transformers, among the most influential neural architectures in modern machine learning, can be rigorously analyzed as discrete approximations to continuous dynamical systems. This perspective formalizes Transformer operations in terms of interacting particle systems, nonlinear transport equations, and operator-theoretic frameworks. It illuminates the connection between deep learning, numerical analysis, and physical dynamical systems, offering a principled mathematical lens for architectural design, stability analysis, and interpretability.

1. Modeling Layer-Wise Propagation as a Discretized Dynamical System

The Transformer update rules—specifically, the sequential application of self-attention and feed-forward layers with residual connections—admit a precise interpretation as numerical integration schemes for ordinary differential equations (ODEs) in a multi-particle context (Lu et al., 2019, Fein-Ashley, 8 Feb 2025, Castin et al., 30 Jan 2025). Each token (word) is viewed as a “particle” whose high-dimensional feature representation $x_i(t)$ evolves across the network’s depth according to

$$\frac{d x_i(t)}{d t} = F\big(x_i(t), [x_1(t),...,x_n(t)], t\big) + G\big(x_i(t), t\big),$$

where $F$ models the interactions (diffusion, via self-attention) among tokens and $G$ represents point-wise (convection-type) transformations from the FFN. The layer updates are mapped to a Lie-Trotter splitting (Euler’s method): $$\begin{aligned} \tilde{x}_i(t) &= x_i(t) + \gamma F\left(x_i(t), [x_1(t),...,x_n(t)], t\right) \ x_i(t+\gamma) &= \tilde{x}_i(t) + \gamma G\left(\tilde{x}_i(t), t\right). \end{aligned}$$ Increasing the number of layers corresponds to refining the time discretization $\gamma\to0$, leading to convergence of the discrete sequence of representations to a solution of the underlying ODE, assuming Lipschitz continuity of $F$ and $G$ (Fein-Ashley, 8 Feb 2025). In the continuum limit, representations propagate smoothly through the network along the ODE’s flow.

The continuous-time ODE perspective also motivates the use of more accurate splitting schemes, such as Strang-Marchuk splitting (Lu et al., 2019), where the FFN is applied in two half-steps sandwiching self-attention—leading to architectures like Macaron Net with empirically observed lower truncation error and stronger downstream performance.

2. Transformers as Mean-Field Particle Systems and Transport PDEs

By lifting the finite set of token representations to a probability measure $\mu$ on $\mathbb{R}^d$, one obtains a mean-field dynamical description. The evolution of the empirical measure $\mu_n = \frac{1}{n}\sum_{i=1}^n\delta_{x_i(t)}$ approaches, as $n\to\infty$, the solution of a Vlasov-type transport equation (“Transformer PDE”) (Castin et al., 30 Jan 2025, Geshkovski et al., 2023): $$\partial_t \mu + \operatorname{div}(\mu \Gamma_\mu) = 0,$$ with the velocity field $\Gamma_\mu$ derived from the (possibly nonlinear) attention mechanism: $$\Gamma_\mu(x) = \frac{\int V y\,e^{Q x\cdot K y} d\mu(y)}{\int e^{Q x\cdot K y} d\mu(y)}$$ (for standard dot-product attention). Variants such as L2, Sinkhorn, and masked attention are encompassed within this framework using corresponding kernels and normalization (Castin et al., 30 Jan 2025).

This PDE-based viewpoint allows rigorous analysis of global existence, well-posedness, and clustering phenomena. Notably, in both finite and infinite population limits, the attention-driven velocity field typically induces attractive interactions that can result in clustering, consensus formation, or anisotropy reduction among tokens (Geshkovski et al., 2023, Castin et al., 30 Jan 2025). Gaussian initial data remain Gaussian under evolution, with explicit ODEs governing mean and covariance flow, revealing rank-deficiency and directional collapse in the covariance as depth increases.

3. Geometry, Energy Functionals, and Operator-Theoretic Interpretation

The deep connections between self-attention dynamics and energy minimization emerge when representing the system as an interacting particle flow on the hypersphere, maintained by layer normalization (Gracyk, 21 Jul 2025, Geshkovski et al., 2023, Tai et al., 5 Oct 2025). The evolution of token embeddings on the unit sphere $S^{d-1}$ obeys: $$\dot{x}_i(t) = P_{x_i(t)}\left(\frac{1}{Z_{\beta,i}} \sum_{j=1}^n e^{\beta \langle x_i(t), x_j(t) \rangle} x_j(t)\right),$$ where $P_{x}$ denotes the projection onto the tangent space at $x$ to enforce the normalization constraint. This induces a flow on the tangent bundle, preserving the geometric structure of the token manifold.

The corresponding mean-field flow is understood as a Wasserstein gradient flow of an energy: $$\mathsf{E}_\beta[\mu] = \frac{1}{2\beta} \iint e^{\beta \langle x, x' \rangle}\, d\mu(x)d\mu(x'),$$ whose maximizers (for attractive attention) are fully clustered measures, while the uniform distribution minimizes repulsive energy (Geshkovski et al., 2023).

Layer normalization within this framework is explained as projection onto sets defined by mean and variance constraints, with explicit closed-form projections derived as part of the flow (Tai et al., 5 Oct 2025). The feedforward network can be interpreted as additional integral or variational operators, and the entire system governed by an integro-differential equation, incorporating time-continuous representations of both token and feature (or “channel”) indices (Tai et al., 5 Oct 2025).

4. Control, Variational, and Optimization Principles

Recasting the Transformer as a controlled dynamical system or as a variational problem provides mathematical clarity and suggests architectural improvements (Gracyk, 21 Jul 2025, Tai et al., 5 Oct 2025, Kan et al., 30 Jan 2025). The evolution of token trajectories is shown to satisfy a Lagrangian or energy functional, leading to projected Euler-Lagrange equations on the appropriate manifold. Loss optimization (e.g., minimizing trajectory deviation or total energy) is equivalent to seeking trajectories that satisfy these variational conditions.

Optimal transport regularization in continuous-time Transformer models enforces uniqueness and regularity of ODE solutions, preventing degenerate or unstable dynamics, and guiding hidden states smoothly through the flow (Kan et al., 30 Jan 2025). Training can then be interpreted as an optimal control problem for a structured integro-differential system, leveraging techniques from optimal control and calculus of variations to analyze convergence, regularity, and stability.

5. Implications for Model Design, Stability, and Future Research

The continuous dynamical systems perspective explains empirical phenomena such as oversmoothing, clustering, stability, and error attenuation across layers (Geshkovski et al., 2023, Fein-Ashley, 8 Feb 2025). When the underlying mapping satisfies a one-sided Lipschitz condition with a negative constant, the dynamics are contractive, resulting in exponential decay of perturbations with increasing depth (Fein-Ashley, 8 Feb 2025). This provides a rigorous basis for robustness against noise and motivates the use of adaptive discretization and feedback schemes for accelerated convergence.

Advanced splitting methods (e.g., Strang-Marchuk), continuous-time attention mechanisms using partial differential equations (2505.20666), and explicit geometric constraints (such as on the tangent bundle or through volume preservation (Brantner et al., 2023)) naturally follow from the operator-theoretic framework. Control-based and variational formulations further inform architecture design and principled stability analysis.

The continuous viewpoint also enables researchers to exploit tools from PDE theory, optimal transport, and variational analysis to model, analyze, and generalize Transformer behavior. As research advances, this framework facilitates the unification of sequential deep learning with foundational mathematical theories, leading to more interpretable, robust, and efficient models.

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References:

• (Lu et al., 2019, Geshkovski et al., 2023, Castin et al., 30 Jan 2025, Gracyk, 21 Jul 2025, Fein-Ashley, 8 Feb 2025, Tai et al., 5 Oct 2025, Kan et al., 30 Jan 2025, Brantner et al., 2023, 2505.20666)