Spectral-submanifold latent dynamics

1. Integration point: use a GRU, continuous-time RNN, or latent neural ODE with hidden state $x\in\mathbb R^n$. Add an encoder into $d=2$ or $4$ modal coordinates, a decoder $W_\theta(z)=V_\theta z+H_\theta(z)$, and a reduced vector field $R_\phi(z)$. Use $z$, rather than $x$, for long-horizon rollout. Estimate the hidden Jacobian $J$ at a reference state with automatic differentiation and initialize $V_\theta$ from eigenvectors associated with slow stable complex-conjugate eigenvalues. 2. Pseudocode: encode $z=E_\theta(x)$; reconstruct $\hat x=W_\theta(z)$; evaluate $\hat F=F_\theta(\hat x,t)$ and $\hat R=R_\phi(z,t)$; calculate $e_{\mathrm{inv}}=D_zW_\theta(z)\hat R-\hat F$; optimize $L=L_{\mathrm{prediction}}+\alpha\|e_{\mathrm{inv}}\|^2+\beta\|E_\theta(W_\theta(z))-z\|^2$. At inference, integrate $\dot z=R_\phi(z,t)$ and decode at output times. 3. Computed from the paper's mechanism: the graph invariance equation, spectral-subspace initialization, and reduced amplitude/frequency structure. Estimated empirically: $J$, $\lambda$, $\omega$, $c_r$, and $c_i$; fit these coefficients by least squares from latent trajectories after training. 4. First cheap experiment: train on a damped Duffing oscillator or a two-mode nonlinear beam surrogate, comparing a linear projection, a standard autoencoder latent ODE, and the SSM model at equal latent dimension. Sweep initial amplitude and measure frequency from zero crossings, amplitude decay, invariance residual, and 50- to 200-step rollout error. The quantitative prediction is that $\Omega(r)$ is approximately affine in $r^2$ with slope $c_i$, while amplitude follows the cubic-corrected decay equation. Rollout error should rise sharply as an omitted eigenvalue approaches the selected spectral band, when the spectral-gap ratio $\min_{j\notin E}|\operatorname{Re}\lambda_j|/\max_{i\in E}|\operatorname{Re}\lambda_i|$ approaches one.