Identifiability-Gated Latent Dynamics

Use a latent neural ODE or recurrent state-space model. The encoder maps a recent observation window $y_{t-L:t}$ to a latent hidden state $h_t$; the decoder predicts $y_t$, while the transition contains two branches: a known or learned measured-coordinate base $F_o^{obs}(y_t,u_t)$ and a coupling branch $B\Phi(y_t,u_t)$. A separate small MLP may predict latent-state evolution $\dot h=G_\theta(h,y,u)$, but its contribution to the observed derivative must pass through the constrained coupling branch. Pseudocode: (1) encode $h_t$; (2) compute $r_t=\dot y_t-F_o^{obs}(y_t,u_t)$ using finite differences or an automatic-differentiation observation model; (3) compute $\phi_t=\Phi(y_t,u_t)$; (4) update the neural model on prediction loss plus $\|r_t-B\phi_t\|^2$; (5) every $M$ epochs solve the ridge or elastic-net regression for $B$ and zero coefficients whose confidence intervals include zero; (6) reject a symbolic update when $\lambda_{\min}(N^{-1}\Phi^\top\Phi)<\gamma$. Quantities inherited from the paper are the measured/hidden partition and residual decomposition. Estimate derivatives, feature covariance, standard errors, and $\gamma$ empirically using minibatch or held-out trajectories. The first cheap experiment is a Duffing-like partially observed oscillator or CartPole with one hidden velocity, comparing a vanilla latent neural ODE against this model on trajectory prediction. Sweep excitation: constant input, random input, and a frequency/speed sweep. The quantitative prediction is that constant-input data will yield a near-zero minimum Gram eigenvalue and unstable coefficient estimates across seeds, whereas the sweep will increase $\lambda_{\min}$ and sharply reduce coefficient variance; prediction error and coefficient variance should improve primarily once $\lambda_{\min}>\gamma$, not smoothly with training time.