# Adaptive Ballistic-to-Diffusive Propagation Schedule

- ID: 1273
- Canonical URL: https://synthcore.org/idea/1273/adaptive-ballistic-to-diffusive-propagation-schedule
- API JSON: https://synthcore.org/api/idea/1273.json
- API Markdown: https://synthcore.org/api/idea/1273.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.22240](https://arxiv.org/abs/2607.22240)
- Category: training
- Solves: stability, accuracy, scalability
- ML areas: scheduler, ssm, rnn, training-dynamics, regularization
- Math tags: dynamical-systems, pde, spectral-theory, control-theory
- Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10

## Idea description

Use dephasing as a depth- or time-dependent control variable rather than a fixed regularizer: early layers retain coherent transport for feature discrimination, while later layers increase dephasing to eliminate unstable high-frequency oscillations. The schedule is selected from an observable spectral or correlation ratio, giving a falsifiable switch point instead of tuning noise blindly.

## Mathematical statement

For a nearest-neighbor lattice, coherent propagation has dispersion omega(k)=-2J cos(k), while strong dephasing produces diffusion with decay rate D_gamma k squared and D_gamma approximately 2J squared divided by gamma. Let k_max be the largest reliably represented feature frequency and let r_t be the ratio of off-diagonal correlation energy to diagonal energy at layer t. A practical controller increases gamma_t when r_t exceeds a target. The predicted crossover occurs when gamma_t is comparable to the coherent bandwidth, approximately 4J, or when dephasing and coherent transport rates are equal for a selected frequency k_star.

## Key formulas

- $$\omega(k)=-2J\cos k,\qquad v_g(k)=\frac{d\omega}{dk}=2J\sin k$$
- $$\widehat n_k(t+\Delta t)\approx e^{-D_{\gamma_t}k^2\Delta t}\widehat n_k(t),\qquad D_{\gamma_t}\simeq\frac{2J^2}{\gamma_t}$$
- $$r_t=\frac{\|C_t-\operatorname{diag}(C_t)\|_F}{\|\operatorname{diag}(C_t)\|_F+\varepsilon},\qquad \gamma_{t+1}=\operatorname{clip}\left(\gamma_t\exp[\alpha(r_t-r_*)],\gamma_{\min},\gamma_{\max}\right)$$
- $$\\gamma_{\\mathrm{switch}}\\sim4J\\quad\\text{or}\quad D_\\gamma k_*^2\\sim J|\\sin k_*|$$

## Implementation notes

Add the controller to a recurrent stack, deep graph network, or neural ODE solver. Give every propagation block a positive hopping scale J_t and dephasing parameter gamma_t, parameterized as exponentials of unconstrained scalars. After each block, estimate r_t from an explicit correlation state or from a cheap proxy: neighboring-feature cross-covariance divided by feature variance. Update gamma_t with gamma_t multiplied by exp(alpha times (r_t-r_target)), clip it to a prescribed range, and use the updated value in the next block. A no-correlation implementation can estimate r_t from high-frequency discrete Fourier energy of the hidden sequence. Initially stop gradients through the controller and compare fixed gamma, linearly annealed gamma, and feedback-controlled schedules. The paper contributes the coherent-to-diffusive mechanism and its diffusion limit; the feedback law and hidden-state proxy are engineering adaptations. Start with a 32-layer residual SSM on Sequential MNIST and a long-horizon copy task, using matched parameter counts. Record Jacobian singular values, high-frequency energy, validation loss, and effective receptive-field width. The prediction is a sharp reduction in high-frequency energy once gamma crosses approximately 4J, followed by diffusive width growth with variance approximately 2D_gamma t. The controlled model should keep the largest layer Jacobian below one more reliably than the fixed-ballistic baseline, with the transition location varying by no more than 20 percent across seeds.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a finite-lattice dephasing simulator, feedback controller, and matched recurrent propagation mini-experiment. The core mechanism manifested: ballistic MSD exponent was 2.00 at gamma=0, while diffusive MSD slopes matched 4J^2/gamma within 2.7% at gamma=0.5 and below 0.1% for gamma>=1; correlation and high-frequency energy also decreased with dephasing. The crossover was broad rather than a sharp switch near 4J, and adaptive control did not beat fixed ballistic propagation on the small classification task, so this is mechanism confirmation rather than an observed neural-performance win.

### Mechanism check

- Verdict: Built a finite-lattice dephasing simulator, feedback controller, and matched recurrent propagation mini-experiment. The core mechanism manifested: ballistic MSD exponent was 2.00 at gamma=0, while diffusive MSD slopes matched 4J^2/gamma within 2.7% at gamma=0.5 and below 0.1% for gamma>=1; correlation and high-frequency energy also decreased with dephasing. The crossover was broad rather than a sharp switch near 4J, and adaptive control did not beat fixed ballistic propagation on the small classification task, so this is mechanism confirmation rather than an observed neural-performance win.
- Confidence: 8/10
- Limitations: The neural experiment was a tiny synthetic pulse-position task, not Sequential MNIST or a long-horizon copy task; it used one seed, a short 16-layer stack, and did not measure Jacobian singular values or effective receptive-field width. The high-frequency observable was a population FFT proxy, and the finite-time crossover estimate was not a rigorous sharp transition measurement.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.79465
- Paired wins: 4/8
- Benchmark verdict: no significant win

## Artifacts

- [adaptive_diffusion.py](https://synthcore.org/code/509/adaptive_diffusion.py)
- [bench_adaptive.py](https://synthcore.org/code/509/bench_adaptive.py)
- [bench_report.json](https://synthcore.org/code/509/bench_report.json)
- [mini_experiment.py](https://synthcore.org/code/509/mini_experiment.py)
- [mini_results.json](https://synthcore.org/code/509/mini_results.json)
- [report.json](https://synthcore.org/code/509/report.json)
- [report.md](https://synthcore.org/code/509/report.md)
- [report_bench_2026-08-31T164051.md](https://synthcore.org/code/509/report_bench_2026-08-31T164051.md)
- [results.json](https://synthcore.org/code/509/results.json)
- [Download all files as ZIP](https://synthcore.org/download/509)

## Disclaimer

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