Coulomb field corrector for particle-based generator training

Add a corrector between generator sampling and the generator update. For each minibatch, calculate raw particles \(x_i=G_\theta(z_i)\) and target particles \(y_j\), normalize them into a periodic box, and construct differentiable grid histograms \(r\) and \(m\). Solve the discrete Poisson equation by FFT: for each nonzero frequency index \(k\), set \(\widehat u(k)=\widehat{(r-m)}(k)/\lambda_k\), where \(\lambda_k\) is the positive eigenvalue of the chosen discrete negative Laplacian, and set \(\widehat u(0)=0\). Differentiate the interpolated potential to obtain \(F_i=\nabla u(x_i)\). To avoid self-force artifacts, either compute the source field with particle \(i\) omitted, use Gaussian splats wider than one grid cell, or subtract the analytically known diagonal contribution. Define \(\mathcal E_k=\langle r-m,u\rangle\) and choose \(\eta_k=\min(\eta_{max},\gamma\mathcal E_k/(N^{-1}\sum_i\|F_i\|^2+\varepsilon))\). Apply one to three steps \(\tilde x_i=x_i-\eta_kF_i\). In the detached version, train with \(\|G_\theta(z_i)-\operatorname{stopgrad}(\tilde x_i)\|^2\); in the fully differentiable version, backpropagate through the field computation. The Poisson solve, velocity, and Euler step are mathematical transfers; detachment, smoothing, and step clipping are empirical adaptations. First compare ordinary GAN, MMD, and GAN-plus-corrector on 2-D eight-Gaussians and Swiss roll using the same MLP generator and minibatch size. Record pre/post-correction Coulomb energy, mode coverage, nearest-target distance, gradient variance, radius growth, and wall-clock cost. The desired signal is a substantial immediate energy reduction and improved mode coverage at equal generator updates without particle explosion or target memorization.