Stable Modal State-Space Bottleneck

1. Exact integration point: use this module as the latent bottleneck of a mesh-based video predictor, neural PDE surrogate, or graph/SSM forecaster. An MLP trunk receives coordinates $\mathbf{s}$ and outputs $r$ values $\phi_i(\mathbf{s})$. An encoder maps the first observed field frames to complex modal amplitudes $q_i[0]$ and, optionally, slowly varying forcing terms. Parameterize damping as $\sigma_i=\operatorname{softplus}(a_i)+\epsilon$ and frequency as $\omega_{d,i}=\operatorname{softplus}(b_i)$, guaranteeing stable free decay. 2. Pseudocode: evaluate $\Phi_{xi}=\phi_i(x)$ on the training grid; normalize with the quadrature mass matrix $W$; add $\|\Phi^TW\Phi-I\|_F^2$; encode $q[0]$; for each future step compute $q_i[k+1]=\exp((-\sigma_i+j\omega_{d,i})\Delta t)q_i[k]$; decode $\hat u[:,k]=\Phi\operatorname{Re}(q[:,k])$; backpropagate reconstruction plus projection losses. For driven systems, add $q_i[k+1]=\alpha_iq_i[k]+\beta_i^Tf[k]$. 3. Computed from the paper: modal reconstruction, orthonormality, pole parameterization, and projection consistency. Estimated empirically: the number of modes $r$, quadrature weights $W$, encoder initialization, loss weights, and whether the modal basis remains useful under nonlinear or forced data. 4. First cheap experiment: train on snapshots of a 1-D damped Euler-Bernoulli beam or a synthetic sum of damped sinusoids, comparing a GRU/standard latent SSM against this module at equal parameter count. Sweep the unconstrained discrete spectral radius and verify that the modal model has no free-decay divergence while baselines diverge when $\rho(A)>1$. The quantitative prediction is that learned decay envelopes satisfy $\log|q_i[k]|\approx-\sigma_i k\Delta t+c_i$ with fitted slope error below 20%, and that the rollout remains bounded for every learned mode because $|\alpha_i|<1$. Also measure $\|\Phi^TW\Phi-I\|_F$ and MAC; the intended signature is orthogonality error below $10^{-2}$ and substantially slower long-horizon error growth than the unconstrained SSM.