Orientation-Preserving Simplex Deformation Layer

Integrate this as a deformation head on a fixed 2D triangular or 3D tetrahedral mesh. The network receives vertex features and predicts displacements \(u_\theta\); set the proposed positions to \(f'=f_{\mathrm{ref}}+u_\theta\), while boundary vertices can be fixed or parameterized separately. For each oriented simplex, assemble \(J_\sigma\), its determinant \(s_\sigma\), and the barrier \(-\mu\log(s_\sigma/s_{\sigma,0})\). Use the total loss consisting of the task loss, an optional edge-length or conformal distortion term, and the barrier. A practical training step is: (1) compute the proposed displacement; (2) if any \(s_\sigma\leq0\), run a feasibility-restoration interpolation \(f\leftarrow(1-t)f_{\mathrm{old}}+tf'\), shrinking \(t\) until all determinants exceed \(\varepsilon>0\); (3) compute sparse determinant gradients and the generalized Laplacian-style geometry gradient; (4) update network parameters with Adam; (5) optionally apply the same backtracking rule to the predicted displacement before committing it. For a direct optimizer experiment, use \(d=-\nabla_fE\), \(\beta=0.5\), and \(\rho=0.9\). The mathematical quantities computed exactly are simplex determinants, barrier values, and their analytic gradients; the network Jacobian and task gradients are obtained by autodiff. Start with a small triangular-grid autoencoder or mesh-registration MLP on synthetic random smooth deformations, comparing unconstrained MSE training, soft negative-determinant penalties, and this barrier method. Measure inversion rate, minimum signed determinant, registration error, and convergence. Success means zero inverted simplices throughout training with equal or lower final task error, and fewer catastrophic loss spikes than the soft-penalty baseline.