Physics-informed neural networks with multi-stage transfer learning for electromagnetic field prediction in electroslag remelting process

Electroslag remelting (ESR) requires rapid and physically consistent electromagnetic-field prediction for controlling Joule heating, molten-pool morphology, inclusion removal and solidification quality. Conventional finite element and finite volume simulations are accurate but computationally expensive and difficult to reconfigure for new operating conditions, whereas purely data-driven models are unreliable under sparse and noisy industrial measurements. This work develops a physics-informed neural network (PINN) with multi-stage transfer learning for ESR electromagnetic-field prediction. Maxwell-equation residuals, axisymmetric boundary constraints and sparse magnetic-flux-density measurements are embedded in a composite loss. Training proceeds through a 50 Hz baseline model, frequency transfer over 10-50 Hz and geometric parameterization of slag-pool thickness over 0.15-0.21 m. The baseline model predicts magnetic flux density with a mean relative error of 0.16% and a maximum point error of 0.27%, while maintaining a 16% power-balance deviation and less than 13% current-conservation error. Frequency transfer shows that increasing frequency from 10 to 50 Hz raises the slag-pool power fraction from 64% to 85%. Increasing slag-pool thickness from 0.15 to 0.21 m raises this fraction from 62% to 88% and reduces the characteristic ingot heating depth from 0.26 to 0.15 m. Ablation and noise tests confirm the complementary roles of physics and data constraints, with a 5.44% prediction error under 10% training-data noise. The framework provides a fast computational kernel for ESR soft sensing, process optimization and digital-twin development.