{
 "artifacts": null,
 "category": "training",
 "description": "Construct a compact latent basis for a parameter-conditioned neural operator by repeatedly adding the parameter sample with the largest normalized prediction residual. This replaces uniform parameter coverage with adaptive basis growth and supplies a stopping rule for when the latent approximation is sufficiently accurate.",
 "formulas_latex": [
  "$$\\mathbb{V}_{N}=\\operatorname{span}\\{f(\\cdot\\,;\\bm{\\mu}_{i})\\}_{i=1}^{N},\\qquad f_{\\mathrm{rb}}(\\cdot\\,;\\bm{\\mu})=\\sum_{i=1}^{N}c_i(\\bm{\\mu})u_i.$$",
  "$$\\mu_{N+1}=\\underset{\\mu\\in\\mathcal{P}_{\\mathrm{train}}}{\\operatorname{arg\\,max}}\\;\\rho_N(\\mu),\\qquad \\rho_N(\\mu)=\\frac{\\|F_\\theta(y_N(\\mu);\\mu)\\|}{\\max(\\|b(\\mu)\\|,\\varepsilon)}.$$",
  "$$u_{N+1}=\\frac{(I-U_NU_N^{\\top})s(\\mu_{N+1})}{\\|(I-U_NU_N^{\\top})s(\\mu_{N+1})\\|},\\qquad \\text{stop when }\\max_{\\mu\\in\\mathcal{P}_{\\mathrm{train}}}\\rho_N(\\mu)\\leq\\tau.$$",
  "$$J(\\bm{\\mu})=\\left\\|T\\left(\\sum_{i=1}^{N}c_i(\\bm{\\mu})u_i\\right)-T^{\\mathrm{data}}\\right\\|_x^2.$$"
 ],
 "id": 3067,
 "implementation": "(1) Integration point: use this procedure for adaptive data selection and latent-basis construction in a parametric neural operator or differentiable simulator. A snapshot s(mu) may be an encoder embedding, decoder output, hidden rollout state, or full predicted field compressed by randomized SVD. Define F_theta as one-step prediction error on a target batch; if a governing equation is available, concatenate its discretized residual. Do not use training loss alone, because it may miss poorly represented parameter regions.\n\n(2) Pseudocode:\n```text\nP = candidate_parameter_samples\nchoose mu_1; S = [snapshot(mu_1)]; U = orthonormalize(S)\nrepeat:\n  for mu in P:\n    z = snapshot_or_encoder(mu)\n    c = U.T @ z\n    yN = U @ c\n    rho[mu] = norm(residual(theta, yN, mu)) / max(norm(scale(mu)), eps)\n  mu_new = argmax(rho)\n  if rho[mu_new] \u003c= tau: break\n  s = snapshot(mu_new)\n  s = s - U @ (U.T @ s)\n  U = concatenate(U, s / (norm(s)+eps))\n  optionally fine-tune theta on selected high-rho samples\n```\n(3) Orthonormalization, residual scoring, greedy argmax selection, and the threshold stopping rule come from the reduced-basis idea. The snapshot type, residual normalization, candidate pool, and threshold tau must be selected empirically. Compute scores in batches and validate them on a disjoint parameter set. (4) First cheap experiment: use a small Fourier neural operator or MLP neural operator on a 1D parametric advection-diffusion family with two to four parameters. Compare random, Latin-hypercube, and residual-greedy sampling using the same number of expensive training solves. Report test error versus snapshot count, worst-case error over parameters, and cost to reach a target error. Success is a 2x reduction in required snapshots or lower worst-case error at equal data-generation cost; for inverse estimation also compare observable misfit J(mu) and recovered-parameter error.",
 "math_summary": "The paper constructs V_N=span{f(.;mu_i)} from parameterized solution snapshots and selects new snapshots by a residual-based greedy rule. Let U_N=[u_1,...,u_N] be orthonormal basis vectors, c_N(mu) reduced coordinates, and y_N(mu)=U_N c_N(mu) the reduced prediction. Let F_theta(y;mu) be a model or governing-equation residual, b(mu) a scale vector, and epsilon\u003e0 a numerical safeguard. Define rho_N(mu)=||F_theta(y_N(mu);mu)||/max(||b(mu)||,epsilon). Choose mu_{N+1}=argmax_{mu in P_train}rho_N(mu), append its snapshot s(mu_{N+1}), and orthogonalize it with u_{N+1}=(I-U_NU_N^T)s/||(I-U_NU_N^T)s||. Stop when the maximum residual is at most tau. The transferable property is adaptive approximation of a parameterized solution manifold using residual information rather than random sampling alone.",
 "math_tags": [
  "approximation-theory",
  "optimization",
  "linear-algebra",
  "pde"
 ],
 "ml_areas": [
  "world-model",
  "curriculum",
  "data-augmentation",
  "fine-tuning"
 ],
 "paper": {
  "arxiv_id": "2609.02578",
  "arxiv_url": "https://arxiv.org/abs/2609.02578",
  "summary_what_math_gives_to_ml": "The paper combines residual-based greedy reduced-basis construction, offline/online separation of a quadratic nonlocal operator, and constrained residual minimization in reduced coordinates. The transferable asset is the representation of a parameterized quadratic map as a small polynomial in latent coefficients with precomputed basis-pair interactions. This can produce compact quadratic latent operators for neural operators, graph simulators, and world models, while residual-greedy sampling can adaptively allocate expensive training data to difficult regions of parameter space. The strongest first experiments are on parametric PDE surrogates, where full-resolution interaction costs and repeated inverse-problem solves are measurable.",
  "title": "Reduced order model for parametric Boltzmann equation and its application to inverse problems",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "sample-efficiency",
  "scalability"
 ],
 "title": "Residual-greedy latent basis expansion",
 "url": "https://synthcore.org/idea/3067/residual-greedy-latent-basis-expansion",
 "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
   }
  }
 }
}