✓✓ Beats tuned baseline 2026

Post-training Lyapunov/IQC gate for neural feedback

Implementation & benchmark of arXiv:2607.11122 — Implicit Neural Networks as Static Controllers: Certificates and Performance Separation

Usefulness8/10

Difficulty7/10

Novelty7/10

Dynamics Stability Accuracy Generalization Rl Training dynamics Optimizer Ssm Control theory Dynamical systems Optimization Linear algebra Spectral theory

Source paper: Implicit Neural Networks as Static Controllers: Certificates and Performance Separation arXiv:2607.11122 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Train a neural feedback law together with explicit well-posedness barriers, then certify the resulting closed loop using a common quadratic Lyapunov and activation-sector certificate. The controller is deployed only if the certificate proves exponential decay or a discounted quadratic-cost bound, converting training into a falsifiable stability-constrained synthesis procedure.

Formulas

$$x_{t+1}=Ax_t+Bu_t,\quad y_t=Hx_t,\quad z_t=\phi(Wz_t+Vy_t+b),\quad u_t=Rz_t+Sy_t.$$

$$z_i(L_iq_i-z_i)\ge0,\qquad \begin{bmatrix}q\\z\end{bmatrix}^{T}\begin{bmatrix}0&\Lambda/2\\\Lambda/2&-I\end{bmatrix}\begin{bmatrix}q\\z\end{bmatrix}\ge0,\quad \Lambda=\operatorname{diag}(L_i).$$

$$V(x)=x^TPx,\quad P\succ0,\qquad V(x_{t+1})-V(x_t)\le-\epsilon\lVert x_t\rVert_2^2.$$

$$\beta V(x_{t+1})-V(x_t)+x_t^TQx_t+u_t^TRu_t\le0\quad\Longrightarrow\quad J_\beta\le x_0^TPx_0.$$

Mathematical statement

Consider the discrete-time LTI plant x_{t+1}=Ax_t+Bu_t, observation y_t=Hx_t, and implicit controller z_t=phi(Wz_t+Vy_t+b), u_t=Rz_t+Sy_t. Here x is the plant state, z is the controller state, u is the control, and phi is componentwise sector-bounded with 0<=phi_i'(s)<=L_i. For each neuron, z_i(L_i q_i-z_i)>=0, where q=Wz+Vy+b. This is represented by the sector IQC [q;z]^T Pi [q;z]>=0 with Pi=[[0,Lambda/2],[Lambda/2,-I]] and Lambda=diag(L_i). A Lyapunov/IQC certificate searches for P=P^T>0 and nonnegative multipliers lambda_i such that the closed-loop one-step dissipation inequality is negative after subtracting weighted sector inequalities. For stage cost c=x^TQx+u^TRu and discount beta in (0,1), the certificate beta V(x_{t+1})-V(x_t)+c(x_t,u_t)<=0, with V=x^TPx, implies J_beta<=x_0^TPx_0.

Implementation notes

1. Integration point: use the controller as an implicit policy in a small continuous-control benchmark, initially a linear plant such as an inverted-pendulum linearization or a randomly generated unstable LTI system. The controller receives y=Hx and returns u; actuator saturation can be represented by a known bounded activation or included as an additional sector nonlinearity. 2. Pseudocode: initialize W,V,R,S,b; add a barrier loss max(0,q_w-0.9)^2 for q_w=L||W||_2 or rho(L|W|); train the discounted rollout loss plus this barrier; periodically freeze controller parameters, construct the closed-loop interconnection, and solve the IQC/Lyapunov SDP for P and multipliers lambda>=0. Continue training only when the SDP is feasible or penalize its negative-eigenvalue violation. After training, independently recompute the SDP and report the margin epsilon. Deploy u only after checking the fixed-point residual and actuator limits. 3. Computed from the paper: the implicit controller interconnection, sector IQC, Lyapunov inequality, and discounted-cost implication. Estimated empirically: activation slope bounds, spectral norms, SDP feasibility, certificate margin, and rollout cost. 4. First cheap experiment: compare an unconstrained MLP policy, a contraction-constrained implicit policy, and the certificate-gated policy on a two- or four-dimensional LTI plant with Q=I, R=0.1I, and beta=0.99. Sweep controller gain and record certified epsilon, rollout divergence rate, and discounted cost. The prediction is that feasible certificates imply geometric state decay, while crossing the SDP feasibility boundary produces unstable or high-cost rollouts; the bound x_0^TPx_0 should exceed measured discounted cost on every tested initial condition.

Verification

Beats tuned baseline

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significant win ✓

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

Stage 1 — Mechanism check agent confidence 8/10

Built a dependency-free scalar implicit tanh controller experiment with sector-IQC verification, a bounded common-quadratic Lyapunov/IQC gate, fixed-point iteration sweeps, and discounted rollouts. The sector inequality held numerically with minimum 0.0, observed fixed-point rates matched predicted |w|, and certificate feasibility switched between w=0.72 and w=0.73, close to the predicted discounted stability boundary 0.7230. The mechanism manifested as a conservative certificate gate: w=0.72 had positive margin 0.0217, while w=0.73 was rejected with margin -0.1256 and higher discounted cost.

Agent confidence: 8/10

Baseline: Ungated implicit tanh sweep: w=0.73 and 0.80 remained bounded in the finite 300-step saturated-tanh rollout, but had discounted costs 0.3104 and 2.0121 and no certificate guarantee.
Idea: IQC/Lyapunov gate: feasible through w=0.72 and infeasible from w=0.73 on the tested grid; predicted boundary 0.7222 for the undiscounted pole and 0.7230 for the discounted pole, observed cutoff [0.72,0.73]. Sector minimum was 0.0; at w=0.72 the observed fixed-point rate was 0.7199999 with LMI margin 0.0217.

Limitations: Only a scalar hand-selected LTI plant and tanh controller were tested. No neural training loop, policy optimization, multidimensional SDP solver, noise or disturbance analysis, actuator saturation certification, or randomized initial-condition validation was performed. The certificate search is a bounded grid approximation rather than a production SDP solver.

How to run: python3 run_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.11122) Sep 2, 2026

Beats the tuned baseline by 62.5% (p=0.0081, wins 8 / 8 paired seeds; metric: lower is better).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0004
Idea mean: 0.0002
Effect (Δ): -0.0003 (−62.5%; negative = idea better)
Wins: 8 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±32.6%
Mechanism: Confirmed ✓
Practical effect: Helps
Baseline tuning: swept over 3 configs

Limitations:

The experiment uses a spectral-norm barrier/projection as a practical Lyapunov/IQC-inspired gate rather than solving a full multidimensional SDP for the nonlinear GRU. It tests one registered dynamics dataset, one shared rnn_small architecture, 20 epochs, and learning-rate sweep only; robustness to disturbances, actuator saturation, explicit common quadratic P, and longer rollouts was not evaluated.

How to run: python3 stage2_bench.py

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

Stage 2 — Benchmark Aug 30, 2026 latest Worked ✓

Agent confidence: 9/10

The idea won on the structurally matched registered dynamics track using the shared rnn_small architecture and equal training budget. Best baseline MSE was 0.00040614 and best gated-idea MSE was 0.00015247; all eight paired differences favored the idea, with permutation p=0.0081, satisfying the harness criterion for a significant win. The mechanism signature was confirmed: the gated recurrent hidden-to-hidden norm averaged 0.719997 versus 3.2214 for baseline.

Baseline: Registered dynamics/rnn_small baseline, tuned lr=0.01: mean test MSE 0.0004061428881, std 0.0001787820253, per-seed [0.0003163953, 0.0003539441, 0.0003631329, 0.0008530545, 0.0004033911, 0.0002051939, 0.0004021165, 0.0003519148].
Delta mean: -0.0002536749762 (idea lower/better; all 8 seeds won)
Idea: Registered dynamics/rnn_small gated training, best lr=0.01: mean test MSE 0.0001524679119, std 0.0000333199003, per-seed [0.0001174607, 0.0001315543, 0.0001559154, 0.0002298424, 0.0001457705, 0.0001197860, 0.0001586169, 0.0001607971].
Mechanism signature: confirmed=true; baseline mean recurrent hidden-to-hidden spectral norm 3.2214289, idea mean 0.7199972, target gate 0.72.
P value: 0.0081

Benchmark result