△ Mechanism confirmed, baseline not beaten 2026

Lyapunov-sign-preserving neural time stepping

Implementation & benchmark of arXiv:2608.29149 — Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

Usefulness8/10

Difficulty5/10

Novelty7/10

Dynamics Stability Accuracy Ssm Rnn World model Diffusion sampling Dynamical systems Stochastic processes Numerical analysis Linear algebra

Source paper: Random attractors and almost-sure stability under discretization of a stochastic autoparametric system arXiv:2608.29149 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Equip a stochastic neural ODE or recurrent state-space model with a step-size controller that explicitly checks whether the discrete-time Lyapunov exponent has the same sign as the continuous-time exponent estimate. If discretization changes an attracting mode into an expanding one, reduce the step size or use a higher-order or semi-implicit update rather than trusting ordinary Euler integration.

Formulas

$$dz_t=f_\theta(z_t,t)\,dt+G_\theta(z_t,t)\,dW_t,$$

$$\lambda=\limsup_{T\to\infty}\frac{1}{T}\log\left\|D\varphi(T,\omega,z_0^\star)\right\|,\qquad \lambda_h=\limsup_{N\to\infty}\frac{1}{Nh}\log\left\|D\Phi_h^N(\omega,z_0^\star)\right\|,$$

$$\widehat\lambda_h=\frac{1}{Nh}\sum_{k=0}^{N-1}\log\left\|J_kq_k\right\|,\qquad q_{k+1}=\frac{J_kq_k}{\|J_kq_k\|},$$

$$|\lambda_h-\lambda|\le Ch^p\quad\Longrightarrow\quad \operatorname{sign}(\lambda_h)=\operatorname{sign}(\lambda)\ \text{if}\ h<\left(\frac{|\lambda|}{C}\right)^{1/p}.$$

Mathematical statement

Let $z_t$ be a neural latent state driven by random input or Wiener noise, with continuous dynamics $dz_t=f_\theta(z_t,t)dt+G_\theta(z_t,t)dW_t$, where $f_\theta$ is the drift, $G_\theta$ is the diffusion matrix, $W_t$ is Brownian motion, and $\theta$ are network parameters. Let $\Phi_h(\omega,z)$ be a one-step numerical cocycle with step size $h$ and noise realization $\omega$. For a reference invariant trajectory $z_t^\star$, define the continuous top Lyapunov exponent $\lambda=\limsup_{T\to\infty}T^{-1}\log\|D\varphi(T,\omega,z_0^\star)\|$ and the discrete exponent $\lambda_h=\limsup_{N\to\infty}(Nh)^{-1}\log\|D\Phi_h^N(\omega,z_0^\star)\|$. If the discretization has exponent error $|\lambda_h-\lambda|\le Ch^p$ for order $p$, then the stability sign is preserved whenever $h<( |\lambda|/C)^{1/p}$.

Implementation notes

Use this as the solver and safety controller for a stochastic neural ODE, latent world model, or continuous-time RNN. The network supplies $f_\theta$ and $G_\theta$; initially leave the training objective unchanged and expose the integration step $h$ as a runtime control variable. Implement Euler-Maruyama and a second, more stable update such as a semi-implicit drift step. Initialize a normalized random probe $q$. For each transition compute $z_{k+1}=\Phi_h(\omega_k,z_k)$, obtain the Jacobian-vector product $v=J_kq$ using automatic differentiation, accumulate $s\leftarrow s+\log(\|v\|+10^{-12})$, and set $q\leftarrow v/(\|v\|+10^{-12})$. Every $M$ steps compute $\widehat\lambda_h=s/(Mh)$. If the estimate is close to zero, halve $h$; if a supposedly stable mode has $\widehat\lambda_h>0$, halve $h$ or switch to the stable solver. Jacobian-vector products are computed exactly; the exponent, convergence order $p$, and constant $C$ are estimated empirically using runs at $h$, $h/2$, and $h/4$. First experiment: train a two-dimensional stochastic neural ODE on noisy damped-oscillator rollouts, compare Euler-Maruyama with the controlled solver, and sweep $h$. The prediction is that the exponent sign becomes invariant under refinement and that $|\widehat\lambda_h-\widehat\lambda_{h/2}|$ scales as $h^p$; any coarse-step sign flip should disappear after halving $h.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a reusable Lyapunov exponent estimator with exact PyTorch JVP support, Euler and semi-implicit stepping, and a controller that halves the step when a nominally stable mode appears expanding. The mechanism manifested: the scalar Euler sign-flip boundary matched h*=2/a within 0.1% across a=0.5,1,2,4, and Euler exponent error showed first-order scaling with successive ratios 2.39, 2.16, and 2.07. Implicit stepping preserved the stable sign in the oscillator sweep, while the controller corrected h=2.5 to h=1.25.

Agent confidence: 9/10

Baseline: Explicit Euler: scalar sign boundary observed at 1.001x the predicted boundary across all tested damping values; oscillator exponent became positive at h=0.1, with lambda_h=+0.0020 despite continuous lambda=-0.2.
Idea: Semi-implicit/controller: implicit oscillator exponent stayed negative for h=0.05, 0.1, 0.2, 0.4, and 0.8; the controller halved h=2.5 after lambda_hat=+0.1622 and stopped at h=1.25 with lambda_hat=-1.1090; exact JVP log error was 0.0.

Limitations: This MVP verifies the mechanism on analytically defined linear deterministic systems and a synthetic PyTorch JVP, not on a trained neural ODE, recurrent model, or genuinely stochastic diffusion process. It does not benchmark wall-clock speed, training loss, memory, or adaptive estimation of the continuous exponent and error constant C.

How to run: python3 lyapunov_controller.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.29149) Sep 2, 2026

Significantly WORSE than the tuned baseline (+4568.4%, p=0.0081, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0021
Idea mean: 0.0983
Effect (Δ): 0.0962 (+4568.4%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±4325.3%
Mechanism: Confirmed ✓
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

Only the registered built-in dynamics track was tested. The intervention used a damped half-step surrogate rather than a full nonlinear implicit solve or adaptive runtime controller; wall-clock, memory, stochastic diffusion, higher-order solvers, and longer-horizon generalization were not tested.

How to run: python3 bench_stage2.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 10/10

The matched Lyapunov-controlled GRU preserved the negative trained-model exponent sign, confirming the proposed mechanism on the dynamics benchmark. However, it significantly worsened standard test MSE: 0.09831 versus 0.002106 for the baseline, with paired delta +0.09620 and permutation p=0.0081; therefore the idea is not a training win.

Baseline: Best lr=0.006; 8-seed test MSE mean=0.0021058194, std=0.0003894366.
Idea: Controlled half-step GRU at shared best lr=0.006; 8-seed test MSE mean=0.0983085380, std=0.1018164212; 0/8 paired wins.
Mechanism signature: map[baseline_discrete_exponent:-0.4939222008 confirmed:true controlled_discrete_exponent:-0.2495538622 observed_sign_preserved:true predicted_sign_preserved:true]

Benchmark result