{
 "artifacts": null,
 "category": "training",
 "description": "Attach a small, online-adapted Taylor-polynomial residual model to a neural dynamics model instead of adapting all network weights. Construct the dictionary from the local Taylor structure, remove monomials forbidden by known joint-odd symmetry, and update its coefficients with recursive least squares. This yields fast adaptation with an interpretable parameter count and a direct test of whether the assumed symmetry is valid.",
 "formulas_latex": [
  "$$\\widehat{F}_{D,k}(x,u)=\\widehat{\\Theta}_{D,k}^{\\top}\\varphi_D^{\\mathrm{FE}}(x,u),$$",
  "$$K_k=\\frac{P_k\\varphi_k}{\\lambda+\\varphi_k^{\\top}P_k\\varphi_k},\\qquad \\widehat{\\Theta}_{k+1}=\\widehat{\\Theta}_k+(x_{k+1}-\\widehat{\\Theta}_k^{\\top}\\varphi_k)K_k^{\\top},$$",
  "$$P_{k+1}=\\lambda^{-1}\\left(P_k-K_k\\varphi_k^{\\top}P_k\\right),\\qquad 0\u003c\\lambda\\le1.$$",
  "$$\\varphi_D^{\\mathrm{odd}}(z)=\\{z^\\alpha:\\ |\\alpha|\\le D,\\ |\\alpha|\\ \\mathrm{odd}\\},$$"
 ],
 "id": 3099,
 "implementation": "Use this as a lightweight adaptation head on a pretrained neural state-space or world model. Freeze the base network $h_\\theta(z)$ and predict $x_{k+1}=h_\\theta(z_k)+R_k^T\\varphi_D^{\\mathrm{FE}}(z_k)$, where $R_k$ is updated online by RLS; alternatively let the polynomial head replace the final transition layer. Build the monomial list once, remove even-total-degree monomials only when the environment has verified joint-odd symmetry, and apply forward-Euler pruning only when the corresponding interaction is physically absent. Pseudocode: initialize $R=0$, $P=\\alpha I$; at each transition compute $\\varphi_k$, residual $e_k=x_{k+1}-h_\\theta(z_k)-R^T\\varphi_k$, gain $K_k=P\\varphi_k/(\\lambda+\\varphi_k^TP\\varphi_k)$, update $R\\leftarrow R+e_kK_k^T$, and update $P\\leftarrow\\lambda^{-1}(P-K_k\\varphi_k^TP)$. The paper provides the Taylor-informed reduced dictionary and online RLS; the neural residual-head construction is the transfer. Compute covariance from $P$ and damp or reject updates when $\\varphi_k^TP\\varphi_k$ is excessively large. The first cheap experiment is a regime-shifted Lorenz-63 or nonlinear-pendulum dataset: pretrain on one parameter range, shift the parameter online, and compare full fine-tuning, unstructured polynomial RLS, and the symmetry/pruned adapter. The prediction is that, when symmetry is correct, even-degree coefficient estimates remain statistically indistinguishable from zero and the reduced adapter reaches a target one-step error with fewer samples. After a regime shift, adaptation error should decrease approximately geometrically at a rate controlled by the forgetting factor $\\lambda$ until reaching a persistent-excitation noise floor.",
 "math_summary": "Let $z=(x,u)$ be the concatenated state-input vector and let $\\varphi_D^{\\mathrm{FE}}(z)$ contain retained monomials through total degree $D$ selected from the forward-Euler/Taylor structure. The identified map is $\\widehat{F}_{D,k}(z)=\\widehat{\\Theta}_{D,k}^{\\top}\\varphi_D^{\\mathrm{FE}}(z)$, where $\\widehat{\\Theta}_{D,k}$ contains one coefficient vector per output dimension. For a measured transition $(z_k,x_{k+1})$, matrix RLS uses gain $K_k$, covariance $P_k$, forgetting factor $0\u003c\\lambda\\le1$, and coefficient residual $e_k$. Joint-odd symmetry retains monomials with odd total degree and removes even-total-degree monomials; additional forward-Euler pruning removes interactions absent from the known local vector field.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "statistics",
  "approximation-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "world-model",
  "ssm",
  "rnn",
  "fine-tuning",
  "training-dynamics"
 ],
 "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": 4,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "sample-efficiency",
  "accuracy",
  "stability"
 ],
 "title": "Taylor-Structured Online Adapter",
 "url": "https://synthcore.org/idea/3099/taylor-structured-online-adapter",
 "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
   }
  }
 }
}