WaveGraphNet: Physics-Consistent Guided-Wave Damage Localization through Coupled Inverse-Forward Graph Learning

Technical Summary: WaveGraphNet

Problem Statement
Guided-wave structural health monitoring (SHM) aims to detect and localize damage in composite materials, such as Carbon Fiber Reinforced Polymer (CFRP) plates, using sparse networks of piezoelectric transducers. A critical challenge in this domain is the "spatial extrapolation" problem: models trained on a limited set of labeled damage locations often perform well on seen regions but fail to generalize to unseen areas of the structure. This limitation arises because inferring defect coordinates from sparse pitch-catch measurements is a weakly constrained inverse problem. Existing machine learning approaches typically treat sensor data as arrays or images, failing to explicitly model the relational topology of the transducer network. Furthermore, standard supervised regression models may predict numerically plausible coordinates that are physically inconsistent with the actual redistribution of wave energy across the sensing network, particularly when extrapolating beyond the training domain.

Methodology: WaveGraphNet
The paper proposes WaveGraphNet, a coupled inverse–forward graph learning framework designed to address the ambiguity of sparse guided-wave localization. The core innovation is the integration of a learned forward model as a physics-consistent regularizer for the inverse localization task.

1. Graph Formulation:

   • The sensing network is explicitly modeled as a graph where transducers are nodes and propagation paths are edges.
   • Inverse Branch ($G_{inv}$): Uses all measured propagation paths ($P$) to map graph-structured spectral descriptors of differential guided-wave responses to a defect coordinate ($\hat{p}$). This branch employs Graph Attention Networks (GAT) to aggregate information from heterogeneous paths.
   • Forward Branch ($G_{fwd}$): Uses a subset of plate-spanning paths ($P_f$) to predict the path-wise energy-deviation patterns ($\Delta E$) induced by a candidate defect location. This branch takes a candidate coordinate as input and outputs the expected energy redistribution.

2. Coupled Training Strategy:
   The framework is trained in three stages to ensure stability and effective learning:

   • Stage I (Inverse Pre-training): The inverse branch is trained solely on the localization loss ($L_{loc}$) to map measurements to coordinates.
   • Stage II (Forward Pre-training): The forward branch is trained to predict energy deviations based on ground-truth coordinates.
   • Stage III (Coupled Optimization): The inverse branch is fine-tuned while the forward branch is frozen. The total loss function ($L_{total}$) combines:
     • Localization Loss: Minimizes the distance between predicted and true coordinates.
     • Forward-Consistency Loss: Penalizes inverse predictions that, when passed through the frozen forward model, produce energy-deviation patterns inconsistent with the measured data.
     • Physics-Correction Loss: Uses the gradient of the forward-consistency loss to guide the inverse prediction toward a physically consistent region.

3. Unified Regression:
   The model handles both damaged and undamaged states within a single regression framework by assigning undamaged samples a fixed "no-damage" target coordinate located outside the admissible plate domain. This eliminates the need for a separate damage detection classifier.

Key Contributions

• Graph-Based Formulation: The authors formulate sparse guided-wave damage localization as a graph-to-coordinate regression problem, explicitly representing transducers as nodes and propagation paths as edges to capture relational structure.
• Coupled Inverse-Forward Framework: They introduce a novel architecture where a learned forward model of path-wise energy deviation acts as a consistency constraint, regularizing the inverse localization to ensure physical plausibility.
• Spatial Extrapolation Evaluation: The study defines and evaluates two spatially held-out train-test splits (Split A and Split B) on the Open Guided Wave (OGW)-1 benchmark. These splits are designed specifically to test extrapolation to unseen damage zones rather than simple interpolation.
• Unified Damage/No-Damage Handling: The framework embeds pristine and damaged states in the same output space, simplifying the inference pipeline.

Results
The framework was evaluated on the OGW-1 SHM Plate benchmark using two spatial hold-out splits of increasing difficulty.

• Extrapolation Performance: The coupled WaveGraphNet achieved the lowest Mean Absolute Error (MAE) in unseen zones compared to non-graph baselines (1D-CNN, LSTM) and graph baselines (GNN-MLP, GAT). In the harder Split B scenario, the coupled model reduced the unseen-zone MAE from 0.383 (GAT) to 0.262.
• False Positive Rate (FPR): The coupled model achieved a 0.0% FPR on held-out undamaged samples in both splits, whereas other models exhibited significant false positives (up to 94.4% for LSTM).
• Ablation: The "Inverse-Only" variant of WaveGraphNet achieved the best performance on seen regions (interpolation) but performed worse than the coupled model on unseen regions. This indicates that the forward-consistency coupling specifically enhances spatial generalization rather than just fitting the training data.

Significance and Claims
The paper claims that WaveGraphNet demonstrates the potential of combining relational graph representations with physics-guided forward consistency constraints to improve spatial generalization in SHM. The authors emphasize that:

• Standard evaluation protocols using random train-test splits may overestimate localization performance; spatially held-out splits reveal significant generalization gaps.
• The coupled inverse-forward approach reduces ambiguity in the inverse problem by ensuring that predicted coordinates are compatible with the measured redistribution of wave energy.
• The framework offers a robust strategy for guided-wave localization under limited spatial coverage, a common constraint in real-world deployment.

The authors modestly frame the study as a controlled experimental case study on the OGW-1 dataset rather than a comprehensive deployment validation. They acknowledge that future work is required to evaluate robustness under environmental variability, sensor drift, and multi-defect scenarios.