Offline quadratic latent operator

(1) Integration point: add a quadratic latent operator after the encoder of a neural operator, graph simulator, or world model. The encoder maps an input field or graph state x to c in R^n. Replace a dense high-resolution pairwise interaction or MLP block with q(c), concatenate [c,q(c)], and pass the result to the decoder or next residual block. This changes both training and inference computation.

(2) Pseudocode:

# Offline
for i in range(n):
  for j in range(n):
    h = expensive_quadratic_operator(u[i], u[j])
    for k in range(n):
      G[k,i,j] = inner_product(u[k], h)
optionally fit low-rank factors A,B to G

# Online
c = encoder(x)
if low_rank:
  t[r] = sum(i,j, B[r,i,j] * c[i] * c[j])
  q[k] = sum(r, A[k,r] * t[r])
else:
  q[k] = sum(i,j, G[k,i,j] * c[i] * c[j])
y = decoder(concat(c, q))

Use symmetry when justified, storing only i<=j and doubling off-diagonal terms. Initialize the quadratic block and its decoder projection near zero so a pretrained baseline is preserved initially. (3) The basis-pair expansion and offline interaction tensor are taken from the paper; the learned encoder, decoder, basis, and factor rank R are engineering choices. Estimate R by randomized SVD or Tucker decomposition of G, retaining 99% of interaction energy. (4) First experiment: train a small Fourier neural operator or graph simulator on a 2D parametric advection-reaction benchmark. Compare a standard MLP interaction, a dense quadratic interaction, and the precomputed reduced operator at matched latent width. Measure forward latency, peak memory, rollout error, and inverse-problem optimization time. Success is at least 2x lower inference cost or memory at matched rollout error, with stable gradients.