Resonance-Aware Stochastic RNN Control

1. Integration point: use a tanh RNN, GRU, or diagonal state-space model with explicit Gaussian noise xi_t in the hidden update. During training, maintain a small ensemble of hidden trajectories under identical inputs but independent noise. Every M optimizer steps, collect one output coordinate, several hidden coordinates, and random projections of the hidden state. 2. Pseudocode: run B noisy trajectories; compute lag covariances C_hat(k)=B^{-1} sum_b (T-k)^{-1} sum_t (y_b,t-y_bar)(y_b,t+k-y_bar); fit m damped complex modes C_hat(k) approximately equal to sum_j a_j z_j^k using matrix-pencil, ESPRIT, or nonlinear least squares; calculate r_hat_j=abs(z_j) and omega_hat_j=arg(z_j); then add lambda times L_res to the task loss. Initially stop gradients through the fitted resonance estimates, or update the penalty only periodically. Estimate Lambda_hat with Jacobian-vector products and tangent-vector renormalization. 3. Computed versus estimated: the transfer-operator interpretation, resonance eigenvalue equation, and correlation expansion are the paper mechanism; finite-dimensional resonance values are estimated from sampled correlations rather than computed exactly. 4. First cheap experiment: train a noisy tanh RNN on a two-sine-wave prediction task and a copy-memory task, comparing ordinary weight decay, recurrent spectral-radius regularization, and resonance regularization. Sweep injected noise amplitude and recurrent gain. The falsifiable prediction is that a narrow output-spectrum peak appears when a fitted r_hat_j crosses approximately 0.95, with correlation decay time close to -1/log(r_hat_j), even when Lambda_hat remains negative. Penalizing r_hat_j above r_max=0.9 should remove that peak and reduce long-horizon error without necessarily changing Lambda_hat. The measured peak frequency should match omega_hat_j divided by 2 pi within about 10 percent.