Hyperuniform Collocation and Minibatch Sampling

1. Integration point: apply this to a coordinate MLP or PINN whose objective is an integral over x in a bounded d-dimensional domain. At every optimizer step, draw collocation points from iid uniform sampling, a randomly shifted lattice, and a hyperuniform approximation. The simplest hyperuniform approximation is a randomly shifted lattice with bounded correlated displacements; a stronger version uses particle relaxation that minimizes low-frequency Fourier power. 2. Pseudocode: initialize points x_j; for T relaxation iterations compute L_HU=sum over nonzero Fourier modes with |k|<k_c of w_k times the squared magnitude of sum_j exp(i k dot x_j); update x_j <- x_j-alpha times gradient_xj L_HU; enforce periodic boundaries or reflection; then evaluate the neural loss and gradient on x_j. Use a fresh random global shift each epoch to avoid fixed-grid bias. 3. Compute S(k) empirically on a Fourier grid, fit its log-log slope near zero, and measure covariance of minibatch gradients. The paper supplies the low-frequency suppression and hyperuniform count-variance targets; the finite-domain sampler and Fourier estimates are approximations. 4. First cheap experiment: train a four-layer width-128 coordinate MLP on a smooth 2D function with 256 points per step, comparing iid, Sobol, shifted-grid, and relaxed hyperuniform points. Measure test error, gradient variance, and Var(N_R) over many windows. The prediction is a low-k slope near at least 2 for q=1, count variance closer to R^(d-1) than R^d, and lower smooth-function gradient variance at equal batch size.