Weak Cartan curvature loss

Integrate this as an auxiliary loss for a neural implicit surface or graph network. Let the network output \(f_\theta(x_1,x_2)\), or more generally an immersion \(X_\theta(u,v)\in\mathbb R^3\). First compute tangent vectors \(X_u,X_v\) and the induced metric \(g_{ab}=\langle X_a,X_b\rangle\). Obtain a numerically stable coframe by Cholesky factorization \(g=LL^T\): if \(du=(du,dv)^T\), use \(\eta=L^{-1}du\), whose components are the orthonormal 1-forms \(\eta^1,\eta^2\). At quadrature points, use autodiff to compute the exterior derivatives of the coframe coefficients. Write \(d\eta^i=a_i\eta^1\wedge\eta^2\), solve \(\omega=-a_1\eta^1-a_2\eta^2\), and form the weak residual for several fixed test functions \(\varphi_m\), such as tensor-product cubic B-splines, compactly supported Gaussian bumps, or low-frequency sine functions. Implement the integration-by-parts version \(-\int d\varphi_m\wedge\omega-\int\varphi_mK_gd\mathrm{vol}_g\), which avoids explicitly differentiating \(\omega\) a second time. Use Monte Carlo or fixed Gauss quadrature and add \(\lambda_C\mathcal L_{\mathrm{weak\text{-}Cartan}}\) to the reconstruction and isometry losses. The quantities computed from the paper are the coframe, the connection reconstruction, and the weak identity; quadrature error, test-function count, and \(\lambda_C\) are empirical choices. Start with a small MLP fitting a sinusoidal or triangulated isometric surface from noisy point samples, using a direct Gauss determinant penalty as baseline. Track pointwise normal error, metric distortion, curvature residual, and training stability under increasing positional noise and Fourier-feature frequency. Success is a lower validation curvature/normal error and fewer exploding gradients at equal training FLOPs; a particularly useful signal is that the weak loss remains stable when the direct \(\det D^2f\) loss becomes noisy.