{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace an expensive nonlinear rollout of a recurrent or neural state-space model by a locally affine rollout whose Jacobian is evaluated once at the current state and then frozen over a short horizon. Use the resulting transition matrix as an explicit stability monitor and optionally penalize or clip its spectral radius, reducing exploding long-horizon predictions without forcing the entire nonlinear network to be globally contractive.",
 "formulas_latex": [
  "$$A_k=\\left.\\frac{\\partial g_{\\theta}(x,u)}{\\partial x}\\right|_{(x_k,u_k)},\\qquad B_k=\\left.\\frac{\\partial g_{\\theta}(x,u)}{\\partial u}\\right|_{(x_k,u_k)},\\qquad c_k=g_{\\theta}(x_k,u_k)-A_kx_k-B_ku_k.$$",
  "$$\\widehat{x}_{k+i+1}=A_k\\widehat{x}_{k+i}+B_k\\widehat{u}_{k+i}+c_k,\\qquad i=0,\\ldots,H-1.$$",
  "$$\\rho(A_k)\u003c1\\ \\Longrightarrow\\ \\|A_k^i\\|\\to0;\\qquad \\mathcal{L}_{\\mathrm{jac}}=\\left[\\max\\left(0,\\rho(A_k)-r\\right)\\right]^2,$$",
  "$$g_{\\theta}(x_k+\\delta x,u_k+\\delta u)=g_{\\theta}(x_k,u_k)+A_k\\delta x+B_k\\delta u+O(\\|\\delta x\\|^2+\\|\\delta u\\|^2).$$"
 ],
 "id": 3098,
 "implementation": "Integrate this at the rollout layer of a recurrent world model, neural state-space model, or SSM that predicts latent states. At the beginning of every training sequence, compute one Jacobian-vector-product-based estimate of $A_k$ and $B_k$ at the observed initial state; construct $c_k$; then use the affine recurrence for an auxiliary horizon loss or for inference-time planning. Pseudocode: (1) evaluate $y=g_\\theta(x_k,u_k)$; (2) obtain $A_k$ and $B_k$ with automatic differentiation, Hutchinson/vector-Jacobian products, or an explicit Jacobian for small latent dimensions; (3) set $c_k=y-A_kx_k-B_ku_k$; (4) repeatedly update $\\hat{x}\\leftarrow A_k\\hat{x}+B_ku+c_k$; (5) minimize prediction loss plus $\\lambda[\\max(0,\\rho(A_k)-r)]^2$. Compute $\\rho(A_k)$ exactly for small states, or estimate it with 10--20 power iterations; use $r\u003c1$, such as $0.98$, as the target. The paper supplies the frozen-affine construction; the contraction penalty and neural rollout use are adaptations. The first cheap experiment is a GRU or MLP state-space model trained on damped nonlinear oscillator or Lorenz-63 sequences, comparing ordinary nonlinear rollout, frozen-affine rollout, and spectral-penalty training. Sweep the learned Jacobian gain and measure the divergence boundary. The quantitative prediction is that affine error grows quadratically with radius around the expansion state, while long-horizon error sharply worsens when measured $\\rho(A_k)$ crosses $1$; the empirical boundary should be within roughly 20% of this predicted threshold.",
 "math_summary": "Let $g_{\\theta}(x,u)$ be a learned one-step state transition, with state $x\\in\\mathbb{R}^{n}$, input $u\\in\\mathbb{R}^{m}$, and parameters $\\theta$. At the current operating point $(x_k,u_k)$, define $A_k=\\partial g_{\\theta}/\\partial x|_{(x_k,u_k)}$, $B_k=\\partial g_{\\theta}/\\partial u|_{(x_k,u_k)}$, and $c_k=g_{\\theta}(x_k,u_k)-A_kx_k-B_ku_k$. The Jacobian-freezing mechanism uses $\\hat{x}_{k+i+1}=A_k\\hat{x}_{k+i}+B_k\\hat{u}_{k+i}+c_k$ over a prediction horizon. For autonomous or fixed-input rollouts, local asymptotic stability is predicted when $\\rho(A_k)\u003c1$, where $\\rho$ is the spectral radius; $\\|A_k\\|_2\u003c1$ is a sufficient contraction condition. The first neglected nonlinear term gives local mismatch of order $O(\\|\\delta x\\|^2+\\|\\delta u\\|^2)$.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "spectral-theory",
  "linear-algebra",
  "approximation-theory"
 ],
 "ml_areas": [
  "world-model",
  "rnn",
  "ssm",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.03348",
  "arxiv_url": "https://arxiv.org/abs/2609.03348",
  "summary_what_math_gives_to_ml": "The paper offers a constructive local-prediction mechanism: identify a Taylor-structured polynomial approximation of a sampled nonlinear flow, then freeze its Jacobian at the current operating point to obtain a computationally cheap affine multi-step predictor. Its transferable asset is the separation between nonlinear model fitting and locally linear prediction, whose spectral properties can be checked at every step. A promising neural-network use is to apply Jacobian-frozen rollouts to recurrent world models or neural state-space models, with an explicit contraction test or penalty. A second transfer is a small, symmetry-informed polynomial adapter updated online by recursive least squares rather than fine-tuning all network weights.",
  "title": "Taylor-Informed Indirect Adaptive Predictive Control Using Jacobian-Frozen Affine Predictors",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Jacobian-Frozen Stable Rollouts",
 "url": "https://synthcore.org/idea/3098/jacobian-frozen-stable-rollouts",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}