Piola-Conditioned Fixed-Reference Neural Operator

(1) Replace the surface or mesh input of a vector neural operator by one fixed reference mesh \(\widehat\Gamma\). For each batch item, provide shape coefficients \(y_1,\ldots,y_K\), compute \(r_{\mathbf y}\), tangential derivatives \(D_T r_{\mathbf y}\), surface Jacobian \(J_{\mathbf y}\), and transformed normals. Apply the Piola map to every vector-valued input feature and inverse-Piola-map the output current or field. Use a mesh graph neural operator, Fourier layer on a parameterized chart, or attention over reference nodes; all batch items then share node and edge indices. (2) Pseudocode: sample \(y\); compute \(r=\hat x+\sum_{j=1}^K y_j\psi_j(\hat x)\); compute \(A=D_T r\) and \(J\); set \(u=J^{-1}A\hat u\); run \(\hat z=F_\theta(\hat x,y,A,J,n,\hat u)\); return \(z=\mathcal P_y\hat z\). For a variational loss, assemble integrands on the reference mesh with transformed features and include the displayed area factor, without an additional geometry-dependent normalization. (3) Compute \(r,A,J\) analytically or by automatic differentiation from the deformation basis; estimate no theorem constants. Monitor whether \(J\) stays positive and reject invalid shapes. (4) First experiment: train a small mesh neural operator on Helmholtz or Maxwell fields over randomly deformed spheres or cubes, comparing this model with a remeshed model and a Cartesian coordinate-conditioned baseline. Measure relative field error, unseen-shape error, batching throughput, and memory. Success is equal accuracy with one shared mesh, at least 2x larger geometry batch size, and smoother error as deformation coefficients vary.