Born-Structured Bilinear Neural Operator

1. Integration point: take a small Fourier neural operator or convolutional operator used for an inverse problem, and replace its pointwise channel-mixing MLP after each spectral convolution with a Born block. The block receives a feature field $u\in\mathbb{R}^{B\times C\times H\times W}$ and computes $N_{3}$ learned basis projections $a_n(u)$ and $b_m(u)$ using two $1\times1$ convolutions or low-rank Fourier projections. 2. Pseudocode: a = proj_a(u) and b = proj_b(u) with shapes [B,N3,H,W]; form pairwise sums and differences across the two feature-index axes; compute prod = ((a[:,None] + b[None,:])2 - (a[:,None] - b[None,:])2) / 4; contract q = einsum('mn,bmnhw->bchw', C, prod); apply an output projection and residual update u_next = u + linear(u) + q. For a quadratic self-interaction, use two independently projected copies of $u$; for a cross-interaction, use the current iterate and the original measurement embedding. 3. The paper-derived components are the bilinear coefficient contraction and exact square-activation multiplication. Estimate the basis projections, coefficients $C$, residual linear operators, and truncation dimension $N_{3}$ by gradient descent. Use real-valued channels for real data, or represent each complex feature as concatenated real and imaginary channels and implement complex multiplication with four real bilinear products. 4. First experiment: train on a $64\times64$ synthetic 2-D inverse-scattering or Poisson inverse-coefficient dataset with additive Gaussian noise, comparing a standard FNO of equal width and parameter count against one or four Born blocks. Measure reconstruction MSE, error versus noise level, parameter count, and peak activation memory. The expected signal is equal-or-better reconstruction at lower parameter count, smoother degradation as noise increases, and faster loss descent at equal FLOPs. Ablate the bilinear block, the square identity, and the truncation dimension $N_{3}$.