Dissipative Neural State-Space Identification

Add a small storage network gφ to the neural state-space model. During each rollout, compute xₖ₊₁, yₖ, the ordinary prediction loss ℓₖ, and Vφ(xₖ). Add λdiss times Ldiss to the prediction objective. Use Vφ(x) = ||gφ(x)||² + δ||x||² with a small fixed δ to ensure nonnegativity. Set dₖ to the squared output error relative to the measured target, or use a separately optimized free-initial-state reference trajectory when comparing fixed and free solutions. Estimate ℓ_hat* as a stop-gradient exponential moving minimum of per-step losses; alternatively maintain an auxiliary free-initial-state rollout and use its average loss. Backpropagate through fθ and Vφ, but periodically freeze θ and evaluate dissipativity residuals on held-out sequences. The paper supplies the storage-function telescoping argument; the storage network and constants are learned or estimated. First experiment: compare an unconstrained MLP state-space model with the regularized model on noisy Duffing and Lorenz-63 data for one-step and 100-step prediction. Plot residuals, cumulative output error, and fixed/free initialization gaps versus horizon. The prediction is that average residual becomes nonpositive, cumulative squared deviation saturates or grows sublinearly, and the fixed/free gap remains bounded. Persistent positive residuals or linear cumulative growth falsify the proposed certificate.