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Spectral Generative Flow Models: A Physics-Inspired Replacement for Vectorized Large Language Models

Andrew Kiruluta

TL;DR

Spectral Generative Flow Models (SGFMs) propose replacing discrete token-based, attention-centric generation with constrained stochastic flows on continuous fields in a multiscale wavelet space. By grounding the dynamics in Navier--Stokes–like transport, enforcing incompressibility, and diffusing in spectral coefficients, SGFMs achieve long-range coherence with sub-quadratic complexity and natural multimodal unification for text and video. The framework combines a physics-informed prior, a multiscale wavelet representation, and diffusion-based uncertainty to deliver a principled, data-efficient alternative to traditional transformers. If realized at scale, SGFMs could enable stable, long-horizon generation across modalities with unified theory-grounded inductive biases, and may benefit from specialized hardware suited to wavelet-domain diffusion and constraint projection.

Abstract

We introduce Spectral Generative Flow Models (SGFMs), a physics-inspired alternative to transformer-based large language models. Instead of representing text or video as sequences of discrete tokens processed by attention, SGFMs treat generation as the evolution of a continuous field governed by constrained stochastic dynamics in a multiscale wavelet basis. This formulation replaces global attention with local operators, spectral projections, and Navier--Stokes-like transport, yielding a generative mechanism grounded in continuity, geometry, and physical structure. Our framework provides three key innovations: (i) a field-theoretic ontology in which text and video are unified as trajectories of a stochastic partial differential equation; (ii) a wavelet-domain representation that induces sparsity, scale separation, and computational efficiency; and (iii) a constrained stochastic flow that enforces stability, coherence, and uncertainty propagation. Together, these components define a generative architecture that departs fundamentally from autoregressive modeling and diffusion-based approaches. SGFMs offer a principled path toward long-range coherence, multimodal generality, and physically structured inductive bias in next-generation generative models.

Spectral Generative Flow Models: A Physics-Inspired Replacement for Vectorized Large Language Models

TL;DR

Spectral Generative Flow Models (SGFMs) propose replacing discrete token-based, attention-centric generation with constrained stochastic flows on continuous fields in a multiscale wavelet space. By grounding the dynamics in Navier--Stokes–like transport, enforcing incompressibility, and diffusing in spectral coefficients, SGFMs achieve long-range coherence with sub-quadratic complexity and natural multimodal unification for text and video. The framework combines a physics-informed prior, a multiscale wavelet representation, and diffusion-based uncertainty to deliver a principled, data-efficient alternative to traditional transformers. If realized at scale, SGFMs could enable stable, long-horizon generation across modalities with unified theory-grounded inductive biases, and may benefit from specialized hardware suited to wavelet-domain diffusion and constraint projection.

Abstract

We introduce Spectral Generative Flow Models (SGFMs), a physics-inspired alternative to transformer-based large language models. Instead of representing text or video as sequences of discrete tokens processed by attention, SGFMs treat generation as the evolution of a continuous field governed by constrained stochastic dynamics in a multiscale wavelet basis. This formulation replaces global attention with local operators, spectral projections, and Navier--Stokes-like transport, yielding a generative mechanism grounded in continuity, geometry, and physical structure. Our framework provides three key innovations: (i) a field-theoretic ontology in which text and video are unified as trajectories of a stochastic partial differential equation; (ii) a wavelet-domain representation that induces sparsity, scale separation, and computational efficiency; and (iii) a constrained stochastic flow that enforces stability, coherence, and uncertainty propagation. Together, these components define a generative architecture that departs fundamentally from autoregressive modeling and diffusion-based approaches. SGFMs offer a principled path toward long-range coherence, multimodal generality, and physically structured inductive bias in next-generation generative models.
Paper Structure (126 sections, 17 theorems, 80 equations, 1 figure)

This paper contains 126 sections, 17 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. From Discrete Token Models to Generative Dynamics
  3. Physical Generative Systems as a Design Template
  4. Generation as Flow in Function Space
  5. Spectral and Multiscale Structure
  6. 2D and 3D Generative Domains
  7. Novelty Relative to State-of-the-Art vLLMs
  8. Ontology.
  9. Interaction mechanism.
  10. Dynamics.
  11. Multimodality.
  12. Inductive bias.
  13. Contributions.
  14. Related Work
  15. Diffusion Models.
  16. ...and 111 more sections

Key Result

Theorem A.1

Under Assumptions A1--A4, for any initial condition $u_0 \in H$, the SPDE admits at least one global weak (martingale) solution

Figures (1)

  • Figure 1: Spectral Generative Flow Model (SGFM). Generation proceeds as constrained diffusion in wavelet coefficient space, followed by physics-guided correction and projection. The same architecture applies to text (2D) and video (3D) domains.

Theorems & Definitions (17)

  • Theorem A.1: Existence of Weak Solutions
  • Theorem A.2: Pathwise Uniqueness in 2D
  • Theorem A.3: Local Well-Posedness in 3D
  • Lemma A.4: Energy Estimate
  • Lemma A.5: Gradient Operator
  • Lemma A.6: Laplacian Operator
  • Lemma A.7: Projection Operator
  • Theorem A.8: Existence and Uniqueness of Diffusion
  • Theorem A.9: Hybrid Well-Posedness
  • Theorem A.10: Stability
  • ...and 7 more