# Disorder-Nonreciprocal Oscillatory RNN

- ID: 1553
- Canonical URL: https://synthcore.org/idea/1553/disorder-nonreciprocal-oscillatory-rnn
- API JSON: https://synthcore.org/api/idea/1553.json
- API Markdown: https://synthcore.org/api/idea/1553.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.28781](https://arxiv.org/abs/2607.28781)
- Category: architecture
- Solves: stability, accuracy, memory
- ML areas: rnn, ssm, training-dynamics
- Math tags: dynamical-systems, bifurcations, linear-algebra, statistical-mechanics
- Ratings: usefulness 8/10; difficulty 5/10; novelty 6/10

## Idea description

Replace a conventional recurrent transition by two coupled hidden channels with equal-and-opposite cross-couplings and a controllable disorder scale. The antisymmetric coupling produces complex recurrent eigenmodes, providing oscillatory memory rather than purely monotone decay, while the disorder parameter controls the real part of the eigenvalues and therefore the stability margin.

## Mathematical statement

Use continuous-time hidden dynamics with h=(h_A,h_B)^T, self-coupling J, non-reciprocal cross-coupling K, and disorder scale sigma>0: F_A(h)=erf((J h_A+K h_B)/(sqrt(2) sigma)) and F_B(h)=erf((J h_B-K h_A)/(sqrt(2) sigma)). The dynamics is dh/dt=F(h)-h. At the zero fixed point, a=sqrt(2/pi) J/sigma and b=sqrt(2/pi) K/sigma, giving Jacobian [[a-1,b],[-b,a-1]]. Its eigenvalues are lambda_plusminus=a-1 plus or minus i b. Continuous-time local stability requires a<1, equivalently sigma>sqrt(2/pi) J. The local oscillation frequency is absolute value of b. For Euler step size Delta t, discrete stability requires the squared modulus |1+Delta t lambda|^2 to be at most one.

## Key formulas

- $$\frac{d h_A}{d t}=-h_A+\operatorname{erf}\!\left(\frac{J h_A+K h_B}{\sqrt{2}\sigma}\right),\qquad \frac{d h_B}{d t}=-h_B+\operatorname{erf}\!\left(\frac{J h_B-K h_A}{\sqrt{2}\sigma}\right)$$
- $$\mathcal{J}_0=\begin{bmatrix}-1+\sqrt{\frac{2}{\pi}}\frac{J}{\sigma}&\sqrt{\frac{2}{\pi}}\frac{K}{\sigma}\\-\sqrt{\frac{2}{\pi}}\frac{K}{\sigma}&-1+\sqrt{\frac{2}{\pi}}\frac{J}{\sigma}\end{bmatrix},\qquad \lambda_\pm=\sqrt{\frac{2}{\pi}}\frac{J}{\sigma}-1\pm i\sqrt{\frac{2}{\pi}}\frac{K}{\sigma}$$
- $$\operatorname{Re}(\lambda_\pm)<0\Longleftrightarrow \sigma>\sigma_c=\sqrt{\frac{2}{\pi}}J$$
- $$|1+\Delta t\lambda_\pm|^2=\left[1+\Delta t\left(\sqrt{\frac{2}{\pi}}\frac{J}{\sigma}-1\right)\right]^2+\left[\Delta t\sqrt{\frac{2}{\pi}}\frac{K}{\sigma}\right]^2$$

## Implementation notes

Integrate this as a recurrent or state-space cell before a standard readout. Split every hidden vector into equal blocks h_A and h_B. In the simplest MVP, use scalar J, K, and sigma shared across coordinates; a stronger version uses diagonal or block-diagonal matrices. Constrain the cross-block matrices to be +K and -K transpose, preserving the non-reciprocal antisymmetric structure. For input u_t, use the following update: z_A=(J h_A+K h_B+B_A u_t)/sqrt(2 sigma squared); z_B=(J h_B-K h_A+B_B u_t)/sqrt(2 sigma squared); d_A=-h_A+erf(z_A); d_B=-h_B+erf(z_B); h_{t+1}=h_t+Delta t [d_A,d_B]. Compute the predicted eigenvalues from J, K, and sigma. Estimate the actual Jacobian around zero or typical hidden states with automatic differentiation, because inputs and learned biases shift the local operating point. The first cheap experiment should compare this cell with a tanh RNN and GRU at matched parameter counts on sequential MNIST and a delayed sine-copy task. Sweep sigma divided by J while holding K fixed. Measure loss, gradient norms, hidden autocorrelation, and finite-horizon Jacobian spectral radius. The predicted transition is at sigma divided by J equal to sqrt(2/pi), approximately 0.798, within 20 percent after accounting for finite step size and nonzero inputs. On the stable side, autocorrelation should oscillate with angular frequency sqrt(2/pi) times absolute K divided by sigma and decay at rate 1-sqrt(2/pi) J divided by sigma. Euler instability should occur when the displayed discrete modulus exceeds one.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: Built the disorder-nonreciprocal oscillatory RNN MVP and verified three quantitative predictions. The continuous stability boundary was predicted at sigma=0.5585 and observed at 0.55; the Euler boundary was predicted at 0.6581 and observed at 0.6589; and the predicted inverse-sigma frequency slope 0.71810 matched the measured 0.71810. The nonlinear FFT was directionally consistent, with predicted 0.120 Hz versus observed 0.100 Hz, confirming oscillatory dynamics, although the delayed-sine learning comparison did not beat tanh.

### Mechanism check

- Verdict: Built the disorder-nonreciprocal oscillatory RNN MVP and verified three quantitative predictions. The continuous stability boundary was predicted at sigma=0.5585 and observed at 0.55; the Euler boundary was predicted at 0.6581 and observed at 0.6589; and the predicted inverse-sigma frequency slope 0.71810 matched the measured 0.71810. The nonlinear FFT was directionally consistent, with predicted 0.120 Hz versus observed 0.100 Hz, confirming oscillatory dynamics, although the delayed-sine learning comparison did not beat tanh.
- Confidence: 9/10
- Limitations: Only a small delayed-sine task was tested; no GRU or sequential MNIST comparison was run. The FFT frequency estimate has coarse finite-window resolution, and the implementation uses scalar shared J/K/sigma rather than learned block-diagonal matrices or input-dependent operating-point Jacobians.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_experiment.py](https://synthcore.org/code/392/bench_experiment.py)
- [bench_report.json](https://synthcore.org/code/392/bench_report.json)
- [experiment.py](https://synthcore.org/code/392/experiment.py)
- [report.md](https://synthcore.org/code/392/report.md)
- [report_bench_2026-08-31T111219.md](https://synthcore.org/code/392/report_bench_2026-08-31T111219.md)
- [results.json](https://synthcore.org/code/392/results.json)
- [Download all files as ZIP](https://synthcore.org/download/392)

## Disclaimer

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