Continuum Transformers

• Continuum transformers are continuous models that reinterpret discrete transformer layers as flows governed by differential equations.
• They integrate higher-order numerical schemes and measure-theoretic attention to facilitate operator learning and spatiotemporal modeling.
• Empirical studies show reduced parameter counts and improved stability, benefiting tasks like time-series interpolation and PDE benchmarking.

A continuum transformer is a generalization of the standard (discrete) transformer architecture in which the layerwise update dynamics, attention mechanisms, or entire input–output mappings are reinterpreted in terms of continuous mathematical structures—dynamical systems, ordinary/partial differential equations, and mappings between infinite-dimensional function or measure spaces. Under this paradigm, transformers are studied and implemented as layers or flows operating in the continuum, permitting robust analysis of their expressivity, stability, numerical properties, and universality, and yielding new classes of architectures for spatiotemporal modeling, operator learning, continual learning, and beyond.

1. Continuous-Time and Continuous-Depth Transformer Models

Classical transformers stack $N$ discrete layers, each composing multi-head self-attention and feed-forward blocks in a residual update. In the continuum transformer framework, this sequence of layers is interpreted as a discretization of a continuous-time flow. Specifically, the depthwise evolution of token representations $X_n$ is modeled as the forward Euler discretization of an ODE:

$\frac{dX(t)}{dt} = F(X(t)), \quad X(0) = X_0,$

where $F$ encapsulates self-attention ($g(X)$) and feed-forward ($h(X)$) sub-mappings. The standard residual update,

$X_{n+1} = X_n + \Delta t\, F(X_n),$

approximates this flow with step size $\Delta t = 1/N$ (Fein-Ashley, 8 Feb 2025, Dutta et al., 2021). The Transformer Flow Approximation Theorem ensures that as $N \to \infty$, the discrete stack tracks the continuous ODE solution with error $O(1/N)$. Under one-sided Lipschitz contractivity ($\lambda < 0$), the flow is exponentially stable, providing quantitative justification for the empirical stability observed in deep transformers (Fein-Ashley, 8 Feb 2025).

Integrating ideas from numerical analysis yields architectural innovations:

• Higher-order integrators (e.g., Runge-Kutta) reduce discretization error.
• Adaptive step sizes allow finer granularity in stiff regions.
• Implicit integration can improve stability for contractive $F$.
• Momentum/acceleration strategies connect to accelerated convergence schemes.

Such continuous-depth models can be implemented by replacing discrete layers with an ODE solver over subblocks (as in OT-Transformer (Kan et al., 30 Jan 2025)), or by parameterizing the ODE vector field directly with small neural networks and integrating adaptively.

2. Continuum Attention and Measure-Theoretic Formulations

Beyond depth, continuum transformers generalize the notion of tokens from discrete embeddings to objects in continuous domains or function spaces. Attention is lifted to an integral operator over (possibly infinite-dimensional) function or measure spaces. For a domain $D \subset \mathbb{R}^d$, self-attention can be written as:

$$(Au)(x) = \int_D V u(y) \cdot \frac{\exp(\langle Q u(x), K u(y)\rangle)}{\int_D \exp(\langle Q u(x), K u(s)\rangle) ds}\, dy,$$

with $Q, K, V$ as learned or parameterized linear maps (Calvello et al., 2024, Geshkovski et al., 2024, Fonseca et al., 2023). Finite transformers correspond to Monte Carlo or quadrature approximations of this operator, and the continuum formulation admits discretization-invariant, mesh-transferable architectures crucial for neural operator learning (Calvello et al., 2024).

From a measure-theoretic perspective, transformer layers are viewed as push-forwards of empirical measures under "in-context maps" $G(\mu, x)$. Under suitable regularity (support-preservation and uniform continuity of the Fréchet derivative), transformers universally approximate all such measure maps (Furuya et al., 30 Sep 2025). In the infinite-depth and mean-field limits, the evolution of the empirical measure is governed by a nonlocal transport PDE (Vlasov equation), establishing a rigorous connection between deep transformers and classical dynamics of interacting particle systems (Furuya et al., 30 Sep 2025, Geshkovski et al., 2024).

3. Operator and Function Space Universality

A key insight of continuum transformers is their universality as mappings between function spaces, providing theoretical guarantees for their application to operator learning—parametrizing solution operators for PDEs, and modeling mappings between spaces of functions (Calvello et al., 2024, Mishra et al., 23 May 2025).

In the operator context, continuum attention employs infinite-dimensional key, query, and value operators (often implemented as Fourier multipliers or integral kernels), and acts on collections of functions $f^{(i)} \in \mathcal{X}$. Transformer forward passes then perform in-context operator learning, which has been proven to correspond exactly to gradient descent in an operator-valued reproducing kernel Hilbert space (RKHS) (Mishra et al., 23 May 2025). In the infinite-depth limit, the operator converges to the Bayes-optimal solution for surrogate modeling tasks under Gaussian process priors.

General continuum transformer neural operators are shown to be universal approximators of continuous operators $G^\dagger: C^s \to C^{s'}$ (or Sobolev maps $W^{s,p}\to W^{s',p'}$), provided only mild nonlinearity and nonlocality in the architecture (Calvello et al., 2024). Function-space patching strategies further improve computational scalability without sacrificing discretization invariance.

4. Continuous-Time, Irregular, and Online Data Modeling

Continuum transformers enable modeling of continuous- or irregular-time series beyond what is possible with discrete positional encodings. Architectures such as ContiFormer (Chen et al., 2024) and CST (Fonseca et al., 2023) integrate neural ODE-based representations of latent trajectories, continuous interpolation of queries, and attention over continuous temporal domains.

ContiFormer lifts keys and values to continuous-time trajectories via neural ODEs; queries are interpolated by splines, and attention scores are integrated over time intervals with quadrature methods. Universality is achieved: any discrete-time attention can be realized as a special case for suitable choices of continuous functions. CST couples integral-attention in physical domains (space-time) with Sobolev-norm regularization to enforce smoothness, supporting interpolation, upsampling, and superior performance in tasks requiring physically plausible outputs (Fonseca et al., 2023).

In the online continual learning domain, transformers adapted with continuum-style streaming attention, replay streams, and parameter updates enable in-context adaptation on non-stationary streams, outperforming prior methods on large-scale benchmarks (Bornschein et al., 2024).

5. Regularization, Optimal Transport, and Expressivity Guarantees

Continuum transformer architectures exploit regularization principles native to their continuous formulations. OT-Transformer (Kan et al., 30 Jan 2025) introduces a kinetic energy penalty, equivalent to an $L^2$ Wasserstein (optimal transport) regularization, promoting straight, unique, and smooth solution trajectories. The loss function,

$$\mathbb{E}_{(x,y)}\left[L(X(T), y; \phi) + \frac{\lambda}{2 d n} \int_0^T \|f(X(t); \theta_{\text{enc}})\|_F^2 \,dt \right],$$

guarantees well-posedness by eliminating degeneracies in the ODE control, ensuring existence and uniqueness via Hamilton–Jacobi–Bellman theory. Empirically, it stabilizes training and improves generalization, reducing parameter counts and variance across tasks (Kan et al., 30 Jan 2025).

Furthermore, measure-theoretic formulations demonstrate that, under moderate conditions (support preservation, regularity), continuum transformers approximate any continuous map between measures (Furuya et al., 30 Sep 2025). These frameworks encompass and generalize finite transformer behavior and provide constructive proofs of universality, up to any prescribed Wasserstein error.

6. Computational Considerations and Empirical Results

Continuum transformer models often trade per-update efficiency for expressivity and stability. Notably,

• TransEvolve (Dutta et al., 2021): collapses $L$ Transformer layers into time-parametrized continuum encoder blocks, precomputing temporal kernels once and reducing parameter count by up to 90% on long sequences.
• OT-Transformer and neural operator variants: achieve strong empirical performance on point cloud, image, and PDE benchmarks, often with dramatically fewer parameters.
• CST: matches or exceeds discrete transformers in video inpainting, time-series interpolation, and scientific computing domains, with guaranteed smoothness in output functions.

Summary tables from the literature (omitted here due to brevity) show that across domains—natural language, vision, operator learning, time-series forecasting—continuum transformers provide better trade-offs in parameter efficiency, generalization, and discretization invariance than their strictly discrete counterparts (Kan et al., 30 Jan 2025, Calvello et al., 2024, Fonseca et al., 2023).

7. Open Problems and Future Directions

Continuum transformers pose— and help clarify—fundamental questions at the intersection of deep learning, functional analysis, and numerical analysis:

• Expressivity boundaries: Characterizing the minimal depth and width required for specific operator or measure-theoretic mappings (Geshkovski et al., 2024).
• Adaptivity: Developing black-box solvers for adaptive depth (as in neural ODEs), and data-driven step sizes to optimize computational and approximation efficiency (Fein-Ashley, 8 Feb 2025, Dutta et al., 2021).
• Scalability: Efficiently implementing integral attention in high dimensions (e.g., via patching or randomized kernel approximations) while retaining discretization invariance (Calvello et al., 2024).
• Generalization theory: Developing precise generalization bounds for infinite-dimensional function or operator learning models.
• Stochasticity and uncertainty quantification: Extending continuum transformer theories to stochastic differential equations and Bayesian formulations (Chen et al., 2024).
• Spatiotemporal and multi-modal extensions: Unifying attention over general topological spaces and modalities, and handling experimental noise and missingness natively (Fonseca et al., 2023, Chen et al., 2024).

The continuum viewpoint illuminates and unifies a wide array of recent transformer innovations and is likely to remain central to advances in operator learning, scientific ML, and sequence modeling.