Weak Tangential-Divergence Curvature Loss

Integrate this at the loss level for an MLP neural implicit representation $u_\theta:\mathbb R^d\to\mathbb R$, such as an occupancy field, signed-distance model, or classifier whose zero level set is the learned boundary. Sample a minibatch $B$ of points near the target level set by mixing labeled points with random perturbations. Choose $K$ fixed affine test fields $v_k(x)=A_kx+b_k$, with random matrices $A_k$ and vectors $b_k$; affine fields make $J_{v_k}=A_k$ available without an extra autodiff graph. For every point x, compute $g=\nabla_xu_\theta(x)$, set $n=g/(\|g\|_2+\varepsilon)$, and evaluate $t_k=\operatorname{tr}(A_k)-n^TA_kn$. Use $c_k(x)=0$ for a minimal tangential-variation prior, or use known boundary-condition targets when the application supplies them. Average the weighted squared residuals, add the standard task loss and Eikonal term, and backpropagate through n and $u_\theta$. The projected divergence is computed from the paper's formula; $K$, the affine-field distribution, sampling weights $w$, epsilon, and lambda are empirical hyperparameters. Start with a 2D noisy-circle or two-moons reconstruction using a small SDF MLP. Compare against task loss plus Eikonal and against Eikonal plus ordinary total variation. Evaluate Chamfer distance, boundary curvature variance, noise robustness, and optimization stability. Success is lower noisy-boundary error and fewer curvature spikes at equal parameter count and compute, with no clean-data accuracy loss.