{
 "artifacts": null,
 "category": "dynamics",
 "description": "Augment a neural dynamics model with a sparse local Taylor residual whose coefficients are updated online by recursive least squares. Use the neural model for global behavior and the Taylor model for short-horizon prediction, where local adaptation can correct payload, friction, actuator, or environment changes without retraining the network.",
 "formulas_latex": [
  "$$\\dot p=v,\\qquad m\\dot v=-mge_3+fRe_3,\\qquad \\dot R=R\\omega^{\\times},\\qquad J\\dot\\omega=-\\omega^{\\times}J\\omega+\\tau.$$",
  "$$\\phi_r(z)=\\left[1,\\{z_i\\}_{|\\alpha|=1},\\{z^{\\alpha}\\}_{2\\leq |\\alpha|\\leq r}\\right]^{\\top},\\qquad z=\\begin{bmatrix}x-x_0\\\\u-u_0\\end{bmatrix}.$$",
  "$$\\hat{x}_{k+1}=\\hat{F}_{\\theta}(x_k,u_k)+W_k\\phi_r(z_k),\\qquad A_k=\\left.\\frac{\\partial\\hat{x}_{k+1}}{\\partial x_k}\\right|_{(x_0,u_0)},\\qquad B_k=\\left.\\frac{\\partial\\hat{x}_{k+1}}{\\partial u_k}\\right|_{(x_0,u_0)}.$$",
  "$$K_k=\\frac{P_{k-1}\\phi_k}{\\lambda_k+\\phi_k^{\\top}P_{k-1}\\phi_k},\\qquad W_k=W_{k-1}+K_k\\left(y_k-W_{k-1}\\phi_k\\right)^{\\top},\\qquad P_k=\\frac{P_{k-1}-K_k\\phi_k^{\\top}P_{k-1}}{\\lambda_k}.$$"
 ],
 "id": 3100,
 "implementation": "1. Integration point: attach this module to a pretrained neural dynamics model used by a Dyna agent, MPC planner, or trajectory optimizer. Initially freeze the neural parameters. Maintain an operating point $(x_0,u_0)$, construct monomials through degree $r=1$, $2$, or $3$, and maintain one RLS covariance and coefficient matrix per output dimension. 2. Pseudocode: observe $(x_k,u_k,x_{k+1})$; compute $z_k=[x_k-x_0,u_k-u_0]$ and $\\phi_k=\\phi_r(z_k)$; calculate residual target $y_k=x_{k+1}-\\hat{F}_{\\theta}(x_k,u_k)$; apply the displayed RLS update; predict with $\\hat{F}_{\\theta}+W_k\\phi_r$; calculate $A_k$ and $B_k$ by automatic differentiation; freeze them for the next $H$ planning steps; recenter $(x_0,u_0)$ when $\\|z_k\\|$ exceeds a threshold. 3. Computed directly: monomials, RLS gains, covariance, and Jacobians. Estimated empirically: the forgetting schedule, covariance initialization, recentering threshold, and useful Taylor order. 4. First cheap experiment: train a small MLP world model on CartPole or Pendulum, then inject an abrupt mass or actuator-scale change. Compare neural-only, order-1, order-2, and order-3 residual models inside an MPC controller with horizon $H=10$. Measure one-step error, ten-step error, rollout divergence, and recovery time. The quantitative prediction is that effective adaptation memory is approximately $N_{\\mathrm{eff}}\\approx1/(1-\\lambda)$: recovery time should scale with this quantity, while stationary variance should increase as $\\lambda$ decreases. Degree two should reduce local multi-step error over degree one only within a neighborhood where its quadratic term exceeds measurement noise; degree three should help only in a smaller, sufficiently excited neighborhood.",
 "math_summary": "For observed transitions $x_{k+1}=F(x_k,u_k)+\\epsilon_k$, define deviations $\\delta x=x-x_0$ and $\\delta u=u-u_0$ around an operating point $(x_0,u_0)$. Let $z=[\\delta x^\\top,\\delta u^\\top]^\\top$ and let $\\phi_r(z)$ contain every monomial of total degree at most $r$. The local residual model is $\\hat{x}_{k+1}=\\hat{F}_{\\theta}(x_k,u_k)+W_k\\phi_r(z_k)$, where $\\hat{F}_{\\theta}$ is a neural predictor and $W_k$ is an online coefficient matrix. For each output row, recursive least squares uses covariance $P_k$, coefficient row $w_k$, forgetting factor $\\lambda_k\\in(0,1]$, gain $K_k$, and residual target $y_k=x_{k+1}-\\hat{F}_{\\theta}(x_k,u_k)$. The paper's horizon approximation freezes the local Jacobian $A_k$ and input matrix $B_k$ during an $H$-step rollout. The mechanism is time-scale separation: fast local adaptation handles mismatch while the neural model remains a slow global prior.",
 "math_tags": [
  "dynamical-systems",
  "control-theory",
  "optimization",
  "linear-algebra"
 ],
 "ml_areas": [
  "world-model",
  "rl",
  "optimizer"
 ],
 "paper": {
  "arxiv_id": "2609.03351",
  "arxiv_url": "https://arxiv.org/abs/2609.03351",
  "summary_what_math_gives_to_ml": "The paper contributes an adaptive predictive-control mechanism based on sparse first-, second-, and third-order Taylor dictionaries, row-wise recursive least-squares identification, variable-rate forgetting, and a Jacobian held fixed over each prediction horizon. Its transferable asset is a cheap local model that adapts rapidly after payload, actuator, or environment changes while preserving a simple finite-horizon predictor. A strong neural-network transfer is to combine a global neural world model with an online Taylor residual model for model-based reinforcement learning or MPC. A second transfer is to identify persistent force or acceleration biases in a separate adaptive trim channel rather than forcing the neural model to relearn them.",
  "title": "Taylor-Informed Predictive Cost Adaptive Control for Quadrotors with Online Gravity-Trim Adaptation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 8
 },
 "solves": [
  "stability",
  "accuracy",
  "sample-efficiency"
 ],
 "title": "Online Taylor Residual World Model",
 "url": "https://synthcore.org/idea/3100/online-taylor-residual-world-model",
 "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
   }
  }
 }
}