# Gaussian-Process Stability-Frontier Expansion

- ID: 2690
- Canonical URL: https://synthcore.org/idea/2690/gaussian-process-stability-frontier-expansion
- API JSON: https://synthcore.org/api/idea/2690.json
- API Markdown: https://synthcore.org/api/idea/2690.md
- Verification status: mechanism_failed
- Source: [arXiv:2608.22661](https://arxiv.org/abs/2608.22661)
- Category: dynamics
- Solves: stability, sample-efficiency, accuracy
- ML areas: rnn, ssm, world-model, optimizer, training-dynamics
- Math tags: dynamical-systems, control-theory, probability, statistics, optimization
- Ratings: usefulness 8/10; difficulty 5/10; novelty 7/10

## Idea description

Train or initialize a Lyapunov certificate for a recurrent, state-space, or neural-ODE model on an inner set, then actively discover a larger stable state envelope instead of assuming that the certificate generalizes out of distribution. A Gaussian process models the signed stability margin or binary long-horizon outcome, and new simulations are selected where posterior uncertainty and proximity to the estimated boundary are both high.

## Mathematical statement

Let f_theta:R^d -> R^d be the discrete-time neural dynamics x_{t+1}=f_theta(x_t), and let V_phi(x)>=0 be a learned Lyapunov function with equilibrium x_star. Define the one-step Lyapunov decrease margin m(x)=V_phi(x)-V_phi(f_theta(x)); positive m(x) indicates decrease. The initial certified set is S_0={x:V_phi(x)<=c_0 and m(x)>=alpha}, where c_0>0 and alpha>0 are thresholds. For an expensive rollout from x, define y(x)=min_{0<=t<T}m(x_t), with x_{t+1}=f_theta(x_t), or use a binary stable label. Fit a Gaussian process to queried labels, obtaining posterior mean mu(x) and standard deviation sigma(x). Select the next query using a boundary acquisition function a(x)=sigma(x)[tau-|mu(x)|]_+, where tau is a small tolerance and [z]_+=max(z,0). Search on an outer Lyapunov shell A_k={x:c_k<=V_phi(x)<=c_{k+1}}. Accept shell expansion only when its empirical failure rate is at most delta. The transferable prediction is a sharp transition in stability probability and GP mean near mu(x)=0, with query complexity governed by boundary complexity rather than state-space volume.

## Key formulas

- $$x_{t+1}=f_{\theta}(x_t),\qquad m(x)=V_{\phi}(x)-V_{\phi}(f_{\theta}(x)),\qquad \mathcal{S}_0=\{x:V_{\phi}(x)\leq c_0,\ m(x)\geq\alpha\}.$$
- $$y(x)=\min_{0\leq t<T}m(x_t),\qquad x_{t+1}=f_{\theta}(x_t),\qquad \mathrm{stable}(x)=\mathbf{1}\left[\max_{0\leq t<T}\|x_t-x^\ast\|\leq r_{\mathrm{safe}}\ \land\ \|x_T-x^\ast\|\leq\varepsilon\right].$$
- $$\mu(x),\sigma^2(x)=\operatorname{GPPosterior}(x\mid\mathcal{D}),\qquad a(x)=\sigma(x)[\tau-|\mu(x)|]_+,\qquad x_{\mathrm{new}}=\arg\max_{x\in\mathcal{A}_k}a(x).$$
- $$\mathcal{A}_k=\{x:c_k\leq V_{\phi}(x)\leq c_{k+1}\},\qquad c_{k+1}\text{ is accepted only if }\frac{1}{q}\sum_{j=1}^{q}\mathbf{1}[y(x_j)>0]\geq 1-\delta.$$

## Implementation notes

Use this as a stability-validation and active-data-collection layer around a recurrent neural network, state-space model, neural ODE discretization, or learned controller. First train the model normally and construct a small positive Lyapunov network V_phi with V_phi(x_star)=0 and V_phi(x)>0. Enforce an initial decrease margin on a small ball by minimizing relu(alpha-(V_phi(x)-V_phi(f_theta(x)))) over sampled states. Initialize a query set with states inside that ball and states on the first outer shell. For every queried state, run the model for T steps, calculate y(x)=min_t[V_phi(x_t)-V_phi(x_{t+1})], and optionally record a binary success label based on terminal error and maximum excursion. Fit an RBF or Matern Gaussian process to these observations. Generate candidate states by sampling random directions and radii, then retain candidates whose Lyapunov value lies in the current shell. Compute the posterior mean and variance and select the candidate maximizing sigma(x)[tau-|mu(x)|]_+. Run its rollout, append the result, and refit the GP. Expand c_k only when q shell samples have failure rate below delta. Compare against uniform and random shell sampling using the same rollout budget. The first experiment should use a tanh RNN on a two-dimensional synthetic stable-system prediction task, T=50, 100 total rollouts, and 10,000 held-out initial states. The predicted signature is that GP queries cluster near mu=0, locate the transition at least three times more efficiently than uniform sampling, and produce a monotonically expanding envelope until the observed shell failure rate reaches delta.

## Verification

- Status: mechanism_failed
- Mechanism evidence: yes
- Mechanism confirmed: no
- Verdict: Built a radial nonlinear discrete-time stability system, exact Lyapunov-margin rollout labels, RBF-GP boundary acquisition, and uniform/random baselines. The mathematical checks passed: the closed-form margin matched direct evaluation to 1.8e-15, predicted instability radii matched sweeps within 1e-4, and margin-vs-parameter scaling had R2=0.9994. However, the GP policy reached the frontier at query 18 versus 17 for uniform/random and had nearly identical mean frontier distance (0.3694 versus 0.3702), so the promised active-query efficiency win was not observed.

### Mechanism check

- Verdict: Built a radial nonlinear discrete-time stability system, exact Lyapunov-margin rollout labels, RBF-GP boundary acquisition, and uniform/random baselines. The mathematical checks passed: the closed-form margin matched direct evaluation to 1.8e-15, predicted instability radii matched sweeps within 1e-4, and margin-vs-parameter scaling had R2=0.9994. However, the GP policy reached the frontier at query 18 versus 17 for uniform/random and had nearly identical mean frontier distance (0.3694 versus 0.3702), so the promised active-query efficiency win was not observed.
- Confidence: 9/10
- Limitations: This is a 2D analytic radial toy system rather than a trained tanh RNN, and it uses noiseless labels, a fixed candidate pool, one seed, and 40 adaptive queries rather than the proposed 100-rollout RNN/10,000-state evaluation. No shell-expansion acceptance study or repeated-seed statistical test was performed.

## Artifacts

- [experiment.py](https://synthcore.org/code/1023/experiment.py)
- [report.md](https://synthcore.org/code/1023/report.md)
- [results.json](https://synthcore.org/code/1023/results.json)
- [Download all files as ZIP](https://synthcore.org/download/1023)

## Disclaimer

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