# Anchored Neural Lyapunov Certificate

- ID: 2450
- Canonical URL: https://synthcore.org/idea/2450/anchored-neural-lyapunov-certificate
- API JSON: https://synthcore.org/api/idea/2450.json
- API Markdown: https://synthcore.org/api/idea/2450.md
- Verification status: failed_benchmark
- Source: [arXiv:2608.17290](https://arxiv.org/abs/2608.17290)
- Category: dynamics
- Solves: stability, generalization
- ML areas: mlp, training-dynamics, regularization, rl
- Math tags: dynamical-systems, control-theory, approximation-theory, optimization
- Ratings: usefulness 8/10; difficulty 5/10; novelty 8/10

## Idea description

Replace an unconstrained scalar MLP certificate with an anchored positive-definite network whose value and gradient are fixed at the equilibrium. Train it so that its Lie derivative along a neural or physical vector field is strictly negative on a prescribed region of attraction. The construction makes stability robust to approximation error: a certificate remains valid whenever the value, gradient, and Lie-derivative errors stay below the target's strict-decrease margin.

## Mathematical statement

Let f: R^n to R^n be a continuously differentiable vector field with equilibrium f(0)=0, and let phi(t,x) be its flow. The domain of attraction is A={x: phi(t,x) tends to 0 as t tends to infinity}. A maximal Lyapunov function U is positive definite, proper on A, and satisfies L_f U(x)=D U(x) f(x)<0 for every x not equal to 0. Use the anchored family V_theta(x)=||x||^2 exp(g_theta(x)), where g_theta is an MLP. This gives V_theta(0)=0, V_theta(x)>0 for x not equal to 0, and D V_theta(0)=0. On a compact annular set K contained in A excluding the origin, assume U(x)>=m_V>0 and -L_f U(x)>=m_D>0. If ||V_theta-U||_{C^1(K)}<=epsilon, then positivity is preserved for epsilon<m_V, and strict decrease is preserved when epsilon times sup_{x in K}||f(x)|| is less than m_D, because |L_f(V_theta-U)| is at most ||D(V_theta-U)|| times ||f||.

## Key formulas

- $\mathcal{L}_{f}V(x)=DV(x)f(x),\qquad \mathcal{A}=\{x\in\mathbb{R}^{n}:\phi(t,x)\to0\ \text{as}\ t\to\infty\}$
- $V_{\theta}(x)=\|x\|^{2}\exp(g_{\theta}(x)),\qquad V_{\theta}(0)=0,\quad V_{\theta}(x)>0\ \text{for }x\neq0,\quad DV_{\theta}(0)=0$
- $\dot V_{\theta}(x)=\mathcal{L}_{f}V_{\theta}(x)=\nabla V_{\theta}(x)^{\mathsf T}f(x)<0\quad\text{for }x\in K$
- $\|V_{\theta}-U\|_{C^{1}(K)}\le\varepsilon,\quad \varepsilon<m_V,\quad \varepsilon\sup_{x\in K}\|f(x)\|<m_D\ \Longrightarrow\ V_{\theta}>0,\ \mathcal{L}_{f}V_{\theta}<0\ \text{on }K$

## Implementation notes

1. Exact integration point: use this network as a certificate for a learned dynamics model f_hat_psi, a neural ODE x_dot=f_hat_psi(x), or a policy-controlled system x_dot=f(x,pi_omega(x)). Give the certificate MLP g_theta the normalized state coordinates used by the dynamics model. Compute V_theta(x)=||x||^2 exp(g_theta(x)); this removes the need to separately enforce V(0)=0 or positivity away from the equilibrium. 2. Pseudocode: sample states x_i from a ball or replay buffer, excluding a small radius around zero; compute V_i=||x_i||^2 exp(g_theta(x_i)), d_i=gradient_x V_theta(x_i) transpose f_psi(x_i) using automatic differentiation, and optimize L=L_task+lambda_D mean([d_i+alpha||x_i||^2]_+^2)+lambda_fit mean[(V_i-U_i)^2]. If no target U is available, omit the fitting term and normalize V using mean[V_i]=1 on a reference annulus. Reject or backtrack a dynamics or controller update when the empirical decrease margin becomes nonpositive. 3. Computed exactly: V_theta, its gradient, and the Lie derivative are obtained by autodiff; f_psi is either the known plant or the current learned model. Estimated empirically: m_V=min_i V_i, m_D=-max_i d_i, F_K=max_i||f(x_i)||, and the C1 approximation error. Use dense validation samples, perturbations, or interval bounds to detect unsampled violations. 4. First cheap experiment: use the two-dimensional nonlinear stable system x_dot_1=-x_1+x_2^2 and x_dot_2=-2x_2, for which a quadratic Lyapunov function can be constructed, and compare an unconstrained scalar MLP against the anchored network on an annulus K={x:rho<=||x||<=R}. Sweep certificate loss weight and network width, then inject parameter noise or fit the certificate from finite samples. The quantitative prediction is a certification boundary: violations should become likely when the measured gradient error exceeds approximately m_D/F_K; below this threshold, the validation fraction with d_i>=0 should remain near zero. For increasing R, the certified sublevel region should expand until the domain-of-attraction boundary, while the minimum Lie-derivative margin should decrease toward zero.

## Verification

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: no_effect
- Verdict: Built an anchored certificate V(x)=||x||² exp(g(x)) with autodiff Lie derivatives and compared it to an unconstrained MLP on the prescribed nonlinear stable system. The exact anchor check gave V(0)=0 and ||DV(0)||=0; the positivity boundary appeared at epsilon/m_V≈1.0, and the adversarial Lie-derivative transition was observed at epsilon≈0.2799 versus the conservative guaranteed bound m_D/F_K≈0.0199. In training, the anchored model had zero nonpositive values and zero Lie-derivative violations, while the baseline had 1.67% nonpositive values, so the mechanism showed a real positivity benefit, although the derivative bound was highly conservative.

### Mechanism check

- Verdict: Built an anchored certificate V(x)=||x||² exp(g(x)) with autodiff Lie derivatives and compared it to an unconstrained MLP on the prescribed nonlinear stable system. The exact anchor check gave V(0)=0 and ||DV(0)||=0; the positivity boundary appeared at epsilon/m_V≈1.0, and the adversarial Lie-derivative transition was observed at epsilon≈0.2799 versus the conservative guaranteed bound m_D/F_K≈0.0199. In training, the anchored model had zero nonpositive values and zero Lie-derivative violations, while the baseline had 1.67% nonpositive values, so the mechanism showed a real positivity benefit, although the derivative bound was highly conservative.
- Confidence: 8/10
- Limitations: Only a two-dimensional known stable plant and one annulus were tested; no learned dynamics, controller, uncertainty, interval certification, sublevel-region expansion, or wall-clock/FLOP comparison was evaluated. The derivative perturbation sweep used a constructed worst-direction perturbation and dense Monte Carlo samples rather than formal interval bounds.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.92075
- Paired wins: 3/8
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_anchored_lyapunov.py](https://synthcore.org/code/845/bench_anchored_lyapunov.py)
- [bench_report.json](https://synthcore.org/code/845/bench_report.json)
- [experiment.py](https://synthcore.org/code/845/experiment.py)
- [report.md](https://synthcore.org/code/845/report.md)
- [report_bench_2026-09-01T122618.md](https://synthcore.org/code/845/report_bench_2026-09-01T122618.md)
- [results.json](https://synthcore.org/code/845/results.json)
- [Download all files as ZIP](https://synthcore.org/download/845)

## Disclaimer

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