Residual-greedy latent basis expansion

(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.

(2) Pseudocode:

P = candidate_parameter_samples
choose mu_1; S = [snapshot(mu_1)]; U = orthonormalize(S)
repeat:
  for mu in P:
    z = snapshot_or_encoder(mu)
    c = U.T @ z
    yN = U @ c
    rho[mu] = norm(residual(theta, yN, mu)) / max(norm(scale(mu)), eps)
  mu_new = argmax(rho)
  if rho[mu_new] <= tau: break
  s = snapshot(mu_new)
  s = s - U @ (U.T @ s)
  U = concatenate(U, s / (norm(s)+eps))
  optionally fine-tune theta on selected high-rho samples

(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.