Hermitian Contraction Regularizer for Complex Neural Dynamics

Use a complex-valued neural ODE, continuous-time state-space model, or complex RNN with hidden dynamics $\dot z=f_\theta(z,x,t)$. Parameterize the metric as $H_\phi(z)=R_\phi(z)^*R_\phi(z)+\epsilon I$, with $\epsilon>0$, or start with a constant diagonal positive metric for the minimum viable experiment. For each minibatch of states, compute $f$, the complex Jacobian $J$ using automatic differentiation, and the directional derivative $\dot H$; form $C=\dot H+J^*H+HJ+2\alpha H$ and add $\lambda_H\max(0,\lambda_{\max}(H^{-1/2}CH^{-1/2}))^2$ to the task loss. Alternatively reduce the ODE solver step when the largest generalized eigenvalue violates the target bound. Jacobians, matrix products, and eigenvalues are computed directly; the invariant region and global metric bounds $m,M$ are estimated from sampled states or interval bounds. The first cheap experiment should train a two-layer complex neural ODE on damped-oscillator trajectory prediction, comparing ordinary MSE training, spectral-Jacobian regularization, and the Hermitian penalty. Perturb initial states and fit the semilog slope of pairwise distances. The prediction is that the observed slope is no greater than $-\alpha$ after transients, within 20% of the sampled certificate. Sweeping penalty strength should reveal a transition when the maximum sampled generalized eigenvalue crosses zero: beyond that boundary perturbations decay, while below it they can persist or amplify.