Harmonic-coordinate neural PDE ansatz

Integrate this as a coordinate front-end for a neural PDE solver. The input is a collocation point x=(x,y); a small complex MLP q_phi(x)=q_R(x)+i q_I(x) produces one complex coordinate, and a second MLP v_theta receives [q_R,q_I] and predicts the physical field u. Use real automatic differentiation: compute q_x, q_y, q_xx, and q_yy separately for the real and imaginary parts. At each batch, form lap_q=q_xx+q_yy and eik=q_x*q_x+q_y*q_y, retaining complex arithmetic as pairs of real tensors. Optimize loss = PDE residual for u plus boundary/data loss plus lambda_h*mean(abs(lap_q)^2) + lambda_e*mean(abs(eik)^2). A practical loop is: x.requires_grad=True; q=q_phi(x); u=v_theta(real(q),imag(q)); r_pde=laplacian(u)-f(x); r_h=laplacian(q); r_e=qx*qx+qy*qy; loss=mean(r_pde**2)+boundary_loss+lambda_h*mean(abs(r_h)**2)+lambda_e*mean(abs(r_e)**2); backpropagate and update phi and theta. Initialize q close to x+i*y by setting the final layer to a fixed affine map plus a small trainable residual, and ramp lambda_h and lambda_e over the first 10 percent of training. The mathematical constraints are exact in the infinite-capacity limit; residual magnitudes are estimated empirically by autodiff. First test on harmonic functions and Poisson problems on the unit square or disk, comparing a standard MLP and a Fourier-feature MLP at equal parameter count and collocation budget. Success means lower Laplace residual and boundary error with fewer collocation points, especially on out-of-sample interior points; also test whether freezing q after pretraining makes v train faster.