Resonant-Mode Observability Regularizer

1. Integration point: use a GRU, neural ODE, or nonlinear state-space model with latent transition zₜ₊₁ = Tθ(zₜ). Maintain a replay buffer of latent trajectories. Every several training steps, estimate K leading Koopman modes with EDMD: form feature vectors from latent states, compute the empirical Gram matrix G and lagged covariance matrix H, solve K_EDMD = G†H, and extract its eigenvectors as approximate modes. Keep real modes directly and split complex modes into real and imaginary channels. 2. Pseudocode: sample K latent states zⱼ; evaluate the mode matrix Aᵢⱼ = fᵢ(zⱼ); compute L_obs = −log(σ_min(A)² + δ) + α||AᵀA − I||²_F; add this to the prediction loss; update θ with the combined loss. For real modes, sample points close to fᵢ = 0 and add a penalty softplus(ε₀ − ||∇fᵢ||)². Near estimated critical points, add softplus(ε₁ − |det(∇²fᵢ)|)². Stop gradients through the EDMD eigendecomposition during the first implementation. 3. Directly computed quantities are the evaluation matrix, singular values, mode correlations, and finite-difference or autodifferentiated derivatives. Estimated quantities are the Koopman modes, eigenvalues, zero-level sets, and critical points. Estimate expansion by measuring the minimum singular value of the Jacobian product D Tⁿ over replay states. 4. First cheap experiment: train a two-dimensional latent GRU on trajectories from the expanding torus map T(x) = 2x modulo 1 and from a smooth perturbed expanding map. Compare ordinary prediction loss against the observability-regularized model using K = 4 modes and four sampled latent states. The predicted signature is a rank transition: when the number of informative sampled states reaches K, σ_min(A) should become positive; duplicated modes or insufficiently excited trajectories should yield σ_min(A) approximately zero. The regularized model should maintain a persistent singular-value gap and distinct estimated eigenvalues. Report whether its median σ_min(A) is at least twice the baseline value and whether long-horizon prediction error decreases.