PI-TransUNet: A Physics-Informed TransUNet Model for Steel Plate Defect Detection in Electromagnetic Tomography

Technical Summary: PI-TransUNet for EMT Steel Plate Defect Detection

Problem Statement
Electromagnetic Tomography (EMT) offers a non-invasive, contactless method for inspecting metallic structures, such as steel plates, by reconstructing internal conductivity or permeability distributions from boundary measurements. However, the EMT inverse problem is inherently ill-posed and nonlinear due to the "soft-field" nature of electromagnetic diffusion. This leads to severe challenges in image reconstruction, including distorted defect boundaries, blurred features, and instability in the presence of measurement noise. While deep learning methods have improved reconstruction speed and accuracy compared to traditional iterative algorithms, purely data-driven models often lack explicit adherence to electromagnetic physical laws, resulting in predictions that may violate physical principles and exhibit poor generalization.

Methodology: PI-TransUNet Framework
The authors propose PI-TransUNet, a physics-informed hybrid imaging framework designed to bridge the gap between data-driven learning and electromagnetic physical constraints. The architecture operates as a closed-loop system comprising two main components:

1. Inverse Reconstruction Generator (TransUNet):

   • Architecture: A modified TransUNet that integrates the local feature extraction capabilities of Convolutional Neural Networks (CNNs) with the global context modeling of Transformers.
   • Input Processing: The model accepts 81-dimensional boundary voltage vectors. Crucially, it incorporates a Spatial Positional Encoding mechanism, concatenating normalized spatial coordinates with voltage data to enhance the network's awareness of the sensor array's topological layout.
   • Feature Extraction: The encoder uses convolutional blocks for hierarchical feature extraction. The bottleneck employs a Transformer module with Multi-Head Self-Attention (MSA) to capture long-range electromagnetic coupling relationships and global dependencies that CNNs typically miss. A channel attention module further refines feature importance.
   • Output: The decoder progressively upsamples features to reconstruct a 2D conductivity distribution image (18×18 grid).

2. Physics-Guided Forward Surrogate Evaluator:

   • To embed physical laws without the computational cost of a full Finite Element Method (FEM) solver during training, a differentiable forward surrogate model based on an improved ResNet is employed.
   • This evaluator takes the reconstructed conductivity image from the generator and predicts the corresponding boundary voltage vector.
   • The parameters of this surrogate are pre-trained on FEM data and frozen during the joint training of the inverse network.

3. Training Strategy and Loss Function:

   • Hybrid Loss: The total loss function combines a Data-Driven Loss (weighted MSE and L1 norm between the reconstructed image and ground truth) and a Physics-Consistency Loss (MSE between the voltage predicted by the forward surrogate and the actual measured voltage).
   • Dynamic Warm-up: To prevent unstable gradients caused by poor initial reconstructions, a dynamic weighting strategy is used. The weight of the physics-consistency loss linearly increases from 0 to 0.5 over the first 40 epochs, allowing the network to first learn spatial defect locations via data loss before strictly enforcing physical constraints.

Key Contributions
The paper outlines four primary contributions:

1. Dual-Network Framework: Establishment of a joint optimization framework that simultaneously solves the forward and inverse problems, ensuring physical plausibility.
2. Hybrid Architecture: Design of a TransUNet-based inverse solver that effectively combines fine-grained local modeling (CNN) with global feature modeling (Transformer) to handle complex defect distributions.
3. Physics-Driven Regularization: Introduction of a forward surrogate evaluator and consistency losses to fuse electromagnetic physical models with data-driven learning, addressing the ill-posed nature of EMT.
4. Comprehensive Validation: Demonstration of the method's efficacy through extensive numerical simulations and laboratory experiments on a custom 9-coil planar EMT platform.

Results
The proposed method was evaluated against traditional algorithms (Linear Back-Projection, Tikhonov regularization) and standard deep learning models (CNN, U-Net) using both simulated and experimental data.

• Ablation Studies: Quantitative results confirmed that both positional encoding and physics-informed constraints individually improve performance. The full PI-TransUNet model achieved the highest Structural Similarity Index (SSIM: 0.9115) and lowest error rates (RMSE: 0.0904, MAE: 0.0276) compared to baseline models.
• Noise-Free Simulation: PI-TransUNet demonstrated superior defect localization and boundary preservation. Correlation Coefficients (CC) exceeded 0.89 for all test cases, significantly outperforming LBP (CC < 0.45) and Tikhonov methods.
• Noise Robustness: Under varying Signal-to-Noise Ratios (SNR), including severe 10 dB noise, the model maintained high fidelity. Even at 10 dB, CC values remained above 0.81, and Image Error (IE) stayed below 0.21, demonstrating strong resilience to measurement noise.
• Experimental Validation: Tests on a physical 9-coil planar EMT system with real steel plate defects confirmed the simulation findings. The proposed method achieved CC values consistently above 0.82 and IE values below 0.15 across various defect configurations, effectively recovering geometric shapes and locations where traditional and standard deep learning methods suffered from artifacts and distortion.

Significance and Claims
The paper claims that PI-TransUNet provides a high-precision, low-artifact, and robust solution for EMT-based non-destructive testing. By integrating physical constraints directly into the learning process, the method overcomes the limitations of purely data-driven models, ensuring that reconstructions adhere to electromagnetic laws while maintaining high accuracy. The authors assert that this approach effectively balances reconstruction accuracy with structural fidelity, making it suitable for complex industrial environments involving multi-defect scenarios and noisy measurements. The work suggests a pathway for future extensions into 3D reconstruction and multi-parameter joint inversion (conductivity and permeability).