Boundary-Equalized Conformal Neural Coordinates

Integrate this into a coordinate MLP or neural implicit surface model whose input is a 2D parameter x=(x1,x2) in a square or disk Omega and whose output is g_theta(x) in R^m. At each training step, sample N interior points x_i and M boundary points b_j; use automatic differentiation to compute the Jacobian J_theta(x_i), the 2-by-2 metric G=J^T J, lambda=sqrt(0.5*trace(G)+epsilon), and phi=log(lambda+epsilon). Compute the conformal penalty ||G-lambda^2 I_2||_F^2. Obtain the Laplacian of phi either by nested autograd Hessians, Delta phi = d^2 phi/dx1^2 + d^2 phi/dx2^2, or by a five-point finite-difference stencil for a cheaper MVP. Compute the boundary penalty as mean((lambda(b_j)-mean_j lambda(b_j))^2). Optimize the weighted sum in the displayed formula jointly with the original reconstruction, rendering, or surface-fitting loss. Stop gradients through lambda only for an ablation, not in the main method. The mathematical quantities are J, G, lambda, phi, and the three regularizers; alpha, beta, and gamma are empirical weights, with epsilon=1e-6. First test on a 2D-to-3D parameterization of a sphere, torus patch, and scanned surface using a 4-layer width-128 SIREN or ReLU MLP, comparing plain task training against conformal-only, harmonic-only, and the full loss. Measure boundary-to-interior scale ratio, angular distortion, reconstruction error, and gradient stability. Success means lower worst-case Jacobian distortion and boundary stretching at equal reconstruction error, or improved held-out interpolation on unseen parameter coordinates.