Boundary-Action Equivariant Neural Operators for Sparse-Sensor PDE Inversion

Physical transformations in sparse-sensor PDE inversion act jointly on the domain, boundary data, sensor coordinates, observations, and solution field. Boundary concatenation and soft equivariance penalties do not enforce one coherent transformation law across this inverse-problem interface. We introduce boundary-action equivariant neural operators whose lifting, latent blocks, and readout are constructed as intertwiners for a physically admissible group, and retain the smallest singular value of the physical sensor-map Jacobian as a separate observability diagnostic rather than conflating symmetry with identifiability. On PDEBench Darcy flow with held-out D4 orbit elements, BC-EQUIVOP obtains 4.7% relative L2 error, 3.9% boundary error, and 2.1% commutation error, compared with 8.7%, 10.6%, and 9.9% for an FNO with boundary channels. Its 16.2-ms latency versus 7.4 ms for that FNO quantifies the computational price of structural coherence. The results support block-level boundary–sensor structure as the operative inductive bias; the measured 2.1% end-to-end residual is not a claim of exact implementation-level equivariance, and observability remains an independent limitation.