Physics-informed deep learning with a Softmin-Fermat time-of-flight loss for ultrasonic inspection of anisotropic welds

Ultrasonic imaging of austenitic welds is degraded by spatially varying anisotropy, which distorts travel times and limits isotropic delay-law focusing. This simulation-based study proposes a weld-specific physics-informed deep learning framework for estimating effective weld-orientation maps from full matrix capture (FMC) data using a differentiable Softmin-Fermat time-of-flight (SF-ToF) loss. Predicted orientations are converted into rotated stiffness tensors and anisotropic slowness maps, and discretized backwall round-trip paths are aggregated through a Softmin approximation of Fermat’s principle to impose delay-consistency during training. The framework was evaluated in a CIVA-MINA multipass austenitic weld simulation, including a held-out welding-procedure shift, using ResUNet, CRNN, and ConvLSTM. Across architectures, SF-ToF regularization reduced orientation tail errors and threshold-exceeding error rates relative to matched data-driven baselines. For ConvLSTM, the selected λ = 5 × 10^4 reduced held-out RMSE from 10.50° to 8.00°, P99 from 47.4° to 29.9°, and PVR from 2.32% to 1.69%. Representative ray-bending TFM and 15-case localization ablations showed that SF-ToF-DL improved focusing consistency relative to isotropic and baseline-DL reconstructions, moving results toward the ground-truth anisotropy reference. These findings support differentiable Fermat-type regularization as a simulation-validated constraint for anisotropic weld inspection.