{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace a dense Koopman autoencoder latent with a sparse code whose active-coordinate support can represent the local dynamical regime or basin. Train reconstruction, latent linear prediction, and multi-step rollout losses jointly; use the learned support as a label-free regime variable and optionally select a local transition matrix for forecasting.",
 "formulas_latex": [
  "$$x_{k+1}=F_{\\Delta t}(x_k),\\qquad x_k\\in\\mathbb{R}^{d_x}.$$",
  "$$z_k=E_\\theta(x_k),\\qquad \\hat{x}_k=D_\\phi(z_k),\\qquad \\hat{z}_{k+h}=K^h z_k.$$",
  "$$\\mathcal{L}=\\|D_\\phi(E_\\theta(x_k))-x_k\\|_2^2+\\alpha\\|E_\\theta(x_{k+1})-K E_\\theta(x_k)\\|_2^2+\\gamma\\frac{1}{H}\\sum_{h=1}^{H}\\|D_\\phi(K^hE_\\theta(x_k))-x_{k+h}\\|_2^2+\\lambda\\|E_\\theta(x_k)\\|_1+\\eta\\|s(E_\\theta(x_{k+1}))-s(E_\\theta(x_k))\\|_1.$$",
  "$$s_j(z)=\\frac{|z_j|}{|z_j|+\\varepsilon},\\qquad m_j(z)=\\mathbf{1}\\{|z_j|\u003e\\tau\\},\\qquad m(z)\\in\\{0,1\\}^{d_z}.$$"
 ],
 "id": 2805,
 "implementation": "Integrate this into a small state-space or world-model network, not into a language model. Use an MLP encoder E_theta mapping state vectors to a latent width d_z such as 32 or 64, an MLP decoder D_phi, and a trainable matrix K. Training batches should contain contiguous windows (x_k,...,x_{k+H}) from trajectories; use H=5 initially and increase to 10 after reconstruction stabilizes. For each window, compute z_k=E_theta(x_k), decode it, compute the one-step latent residual E_theta(x_{k+1})-K z_k, recursively generate K^h z_k, decode every rollout, and add the L1 latent penalty. Compute s(z) with absolute value divided by absolute value plus epsilon, using epsilon=10^{-6}, and add the support-persistence penalty for adjacent states from the same trajectory. Pseudocode: `z=E(x[:,0]); rec=mean((D(z)-x[:,0])^2); pred_lat=0; pred_obs=0; sparse=mean(abs(z)); support=0; for h in 1..H: z=K @ z; pred_lat += mean((z-E(x[:,h]))^2); pred_obs += mean((D(z)-x[:,h])^2); support += mean(abs(s(E(x[:,h]))-s(E(x[:,h-1])))); loss=rec+alpha*pred_lat/H+gamma*pred_obs/H+lambda*sparse+eta*support`. Select tau on validation data to produce an average active fraction of 10 to 30 percent, then cluster binary masks to test basin separation. The first experiment should use a two-basin nonlinear system such as a double-well oscillator or a mixture of two stable systems, with 1,000 training trajectories and held-out trajectories from each basin. Compare against a dense Koopman autoencoder with identical encoder width, parameter count, optimizer, and rollout loss. Success requires lower 20-step state MSE, less long-rollout divergence, and support-mask classification accuracy above the dense baseline on held-out basin-interior states. Include ablations without support persistence and with only one-step loss; the method is supported only if sparse supports remain stable along trajectories and forecasting improves at equal latent width.",
 "math_summary": "The paper models sampled dynamics by the discrete recurrence x_{k+1}=F_{Delta t}(x_k), where x_k is the observed state in R^{d_x} and F_{Delta t} is the unknown time-Delta t flow map. Introduce an encoder E_theta from R^{d_x} to R^{d_z}, a decoder D_phi from R^{d_z} to R^{d_x}, and a latent linear operator K in R^{d_z times d_z}, with sparse latent code z_k=E_theta(x_k). The adapted objective is L=L_rec+alpha L_koop+gamma L_roll+lambda L_sparse+eta L_support. Here L_rec=||D_phi(z_k)-x_k||_2^2, L_koop=||z_{k+1}-K z_k||_2^2, L_roll=(1/H) sum from h=1 to H of ||D_phi(K^h z_k)-x_{k+h}||_2^2, and L_sparse=||z_k||_1. Define the soft support s_j(z)=|z_j|/(|z_j|+epsilon), where epsilon is positive, and encourage support persistence along a trajectory with L_support=||s(z_{k+1})-s(z_k)||_1. At inference, the discrete support is m_j(z)=1 if |z_j| exceeds tau and 0 otherwise, where tau is a validation-selected threshold. The intended mechanism is that different basins use different active coordinate subsets while each subset admits approximately linear evolution under K.",
 "math_tags": [
  "dynamical-systems",
  "linear-algebra",
  "optimization",
  "sparsity"
 ],
 "ml_areas": [
  "ssm",
  "world-model",
  "training-dynamics",
  "representation"
 ],
 "paper": {
  "arxiv_id": "2608.29057",
  "arxiv_url": "https://arxiv.org/abs/2608.29057",
  "summary_what_math_gives_to_ml": "The paper offers a transferable representation-learning principle for systems whose dynamics are only locally linearizable: make the Koopman latent state sparse, so the active-coordinate support becomes an unsupervised regime variable. The useful asset is not merely an L1 penalty, which is standard, but the coupling of sparse supports with a linear latent transition and reconstruction constraint: different basins can occupy different coordinate subspaces instead of being forced into one globally valid linear model. A practical neural implementation is a sparse Koopman autoencoder trained with one-step and multi-step latent prediction losses, followed by support-conditioned diagnostics or local transition operators. The central falsifiable claim is improved long-horizon forecasting and basin identification over a dense-latent Koopman autoencoder at comparable latent width and compute.",
  "title": "Sparse Koopman Autoencoders Identify Local Dynamical Regimes in Multibasin Systems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "accuracy",
  "generalization",
  "stability"
 ],
 "title": "Support-Sparse Koopman World Model",
 "url": "https://synthcore.org/idea/2805/support-sparse-koopman-world-model",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}