Orientation-consistent adaptive tetrahedral neural field

Integrate this as a replacement for a dense voxel feature grid in a coordinate-based 3D neural field, radiance field, occupancy field, or PDE surrogate. (1) Build an initial tetrahedral mesh covering the scene or simulation domain. Assign every global vertex a stable integer key, sort the four keys in each tetrahedron to form its orientation, and canonicalize each edge and triangular face by its sorted vertex tuple. Store shared boundary coefficients once and element-interior coefficients separately. For each query point, use a tetrahedral point-location data structure, compute barycentric coordinates \(\lambda_0,\ldots,\lambda_3\), evaluate the degree-\(p\) polynomial, and optionally pass polynomial features and normalized barycentric coordinates through a shared MLP. (2) Pseudocode: initialize mesh \(\mathcal{T}\), coefficients \(c\), and MLP parameters \(\theta\); for each batch point \(x\), locate \(T(x)\), compute \(\lambda(x)\), evaluate \(f_T(x)\), predict \(\hat y=g_\theta(f_T(x),\lambda(x))\), and backpropagate the task loss. Every \(K\) steps, sample points in each tetrahedron, compute \(r(x_q)\), calculate \(\eta_T\), mark the top \(q\%\) elements, apply red-green refinement, and initialize child coefficients by evaluating the parent polynomial at child nodes. (3) The orientation and shared-face machinery are mathematical components from the paper; the residual definition, threshold \(\tau\), refinement fraction, and degree \(p\) are empirical choices. (4) First experiment: fit synthetic signed-distance fields and a small NeRF scene, comparing against a dense voxel-grid feature model and a hash-grid encoding at matched error. Measure active feature memory, query throughput, convergence, and continuity jumps across shared faces. Success means at least 2x lower feature memory at equal SDF error or PSNR, with no systematic interface discontinuities and faster loss reduction at equal parameter count.