DMDc-Initialized Marginal-Stable Neural SSM

Integrate this at the transition step of a compact recurrent model or diagonal/state-space sequence model. First choose a scalar or low-dimensional observed signal y_t, an input vector u_t, delay length L, and latent dimension d. Build causal delay vectors z_t by stacking the most recent L observations; if the raw observation is multidimensional, concatenate delayed channels and reduce the resulting matrix to d dimensions with truncated SVD. Construct Z_- and Z_+ from adjacent delay states and U_- from aligned inputs. Compute the regularized DMDc estimate [A B]=Z_+[Z_-;U_-]^T(([Z_-;U_-][Z_-;U_-]^T)+alpha I)^{-1}, with small alpha for noisy data. Take the rank-d SVD truncation before fitting if the Hankel matrix is large. Initialize the neural latent state h_0 from the projected first delay vector, then run: h_pred=A h_t+B u_t; r=g_theta(h_t,u_t); h_{t+1}=h_pred+r; yhat_t=readout(h_t). Start with g_theta as a two-layer width-2d MLP and optionally scale its output by a learnable gate initialized to 0. Compute eigendecomposition of A, record eigenvalues, and identify slow modes by minimizing ||lambda|-1| among stable eigenvalues. During training compare three variants: random A, DMDc A without constraints, and DMDc A with spectral projection or a penalty alpha_s sum_j max(0,|lambda_j|-1)^2. What comes directly from the mathematics is the delay lifting, least-squares DMDc initialization, eigenvalue-based mode selection, and stability projection; the residual network, rank choice, and penalty strength are empirical design choices. The first cheap experiment should use a small copy task, adding-machine task, or Lorenz-system forecasting with sparse inputs, followed by a real electricity or battery time-series benchmark. Train all variants at equal parameter count and FLOPs. Measure validation loss versus optimization steps, gradient norm variance, spectral radius, and mean-squared error at horizons 10x and 50x the training rollout. Success means faster early loss descent, no exploding trajectories, and lower long-horizon error than the randomly initialized recurrent baseline.