A Christoffel-constrained neural network for rapid full-focus imaging of welds in thick-walled austenitic stainless steel pipes

In ultrasonic testing of thick-walled austenitic stainless steel pipe welds, the traditional Total Focusing Method (TFM) suffers from low computational efficiency and poor accuracy for deep defects due to strong acoustic anisotropy. To address this, we propose a fast imaging method using a neural network constrained by Christoffel equations, based on the non-uniform grain orientation field (Ogilvy model). This approach uses automatic differentiation to extract the spatial gradient of the travel time field for the slowness vector, which is substituted into the Christoffel eigen-equation to construct a physical loss function. By integrating boundary supervision with dense physical constraints, the framework transforms complex iterative ray-tracing into offline physical learning and efficient forward inference. Validation using CIVA Full Matrix Capture data demonstrates significantly reduced delay-law computation time while maintaining low travel-time prediction errors. Compared to conventional TFM and the Fast Marching Method, this approach effectively overcomes wavefront distortion and energy divergence, showing excellent robustness to parameter perturbations. This study balances physical consistency with inference efficiency, enabling highly efficient, high-precision TFM imaging of thick-walled pipe welds.