# Brjuno Resonance Curriculum

- ID: 1406
- Canonical URL: https://synthcore.org/idea/1406/brjuno-resonance-curriculum
- API JSON: https://synthcore.org/api/idea/1406.json
- API Markdown: https://synthcore.org/api/idea/1406.md
- Verification status: unverified
- Source: [arXiv:2607.25610](https://arxiv.org/abs/2607.25610)
- Category: training
- Solves: stability, accuracy
- ML areas: ssm, training-dynamics, optimizer, world-model
- Math tags: dynamical-systems, harmonic-analysis, optimization, number-theory
- Ratings: usefulness 5/10; difficulty 5/10; novelty 8/10

## Idea description

Train Fourier or state-space neural models by eliminating well-conditioned spectral modes first and retaining near-resonant modes until a later stage. The schedule is determined by the small-divisor geometry of a reference transport vector, with a cumulative Brjuno-like budget controlling how aggressively spectral corrections may be applied. This should prevent rare nearly resonant modes from producing disproportionately large gradients or unstable long-horizon rollouts.

## Mathematical statement

For a periodic representation with integer mode k in Zᵈ, reference frequency or transport vector ω in Rᵈ, and mode amplitude fₖ, the linearized correction has the small-divisor form uₖ = fₖ divided by i(k·ω). Modes satisfying |k·ω| ≤ σₙ|k| are retained at multiscale stage n, while complementary modes are corrected or trained first. The paper defines the resonance cone K⁺σₙ(ω) = {k: |k·ω| ≤ σₙ|k|}. In one dimension, a proxy for arithmetic difficulty is the Brjuno sum B(α) = Σⱼ log(qⱼ₊₁) divided by qⱼ, where qⱼ are continued-fraction denominators of the irrational frequency α. The neural adaptation uses a finite budget Bᴺ = Σₙ wₙ log(1/σₙ), with wₙ proportional to the spectral energy retained at stage n; training updates are damped when Bᴺ exceeds a prescribed stability budget. The mechanism is suppression of amplification by small denominators, not ordinary frequency dropout.

## Key formulas

- K⁺σₙ(ω) = { k ∈ Zᵈ : |k·ω| ≤ σₙ |k| }
- uₖ = fₖ / (i k·ω), for k not in K⁺σₙ(ω)
- B(α) = Σⱼ≥0 log(qⱼ₊₁) / qⱼ,   Bᴺ = Σₙ₌₀ᴺ⁻¹ wₙ log(1/σₙ)

## Implementation notes

Use a Fourier neural operator, periodic convolutional model, or diagonal state-space model whose hidden states are represented in Fourier modes. First estimate a reference transport vector ω: for synthetic advection data it is known; for real data fit it from phase shifts by least squares, and stop gradients through this estimate during the first experiment. At every training step, compute rₖ = |k·ω|/(|k| + ε) for every retained mode. Maintain geometrically shrinking thresholds σₙ = σ₀·2⁻ⁿ. Pseudocode: compute the FFT of the residual; mark a mode resonant when rₖ ≤ σₙ; multiply nonresonant updates by 1 and resonant-mode updates by gₙ = min(1, C/(1 + Bᴺ)); add the next resonance shell only when the exponentially weighted nonresonant residual falls below a tolerance. For a direct homological preconditioner, use correctionₖ = residualₖ/(i(k·ω) + δ) only on nonresonant modes and leave resonant modes unpreconditioned. The cone definition and small-divisor correction come from the paper; the finite budget, damping rule, and shell schedule are neural adaptations. The first cheap test should use a two-dimensional periodic advection equation or Kuramoto-style transport, comparing a Fourier neural operator with Adam against the resonance curriculum at equal parameter count. Sweep σ₀ and learning rate. The predicted signature is a sharp rise in gradient norm and rollout error when the active spectrum first contains modes with rₖ < σ; curriculum training should move this boundary to smaller rₖ. Runs with cumulative budget Bᴺ above a fixed threshold should show increasing spectral correction variance. Across seeds, the instability boundary should be reproducible within roughly 20%, while the curriculum should reduce variance specifically near the smallest-divisor modes rather than uniformly reducing all gradients.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.