Full Waveform Inversion (FWI) has emerged as a powerful method in ultrasonic testing for reconstructing internal material distribution. By utilizing gradient-based optimization to minimize the discrepancy between simulated wave fields and sparse experimental measurements, FWI can resolve high-resolution features of the material distribution. However, the inversion process is inherently ill-posed, often resulting in reconstructions plagued by non-physical artifacts when traditional regularization is absent. Neural network-based parameterizations have recently demonstrated superior performance in regularizing these inverse problems[1]. We introduced a transfer learning strategy in [2] to achieve faster convergence for the FWI task, where a Convolutional Neural Network (CNN) is pre-trained to map the adjoint gradient from the first iteration of a classical FWI to the corresponding damage distribution on a 2D dataset. This strategy of pre-training the neural network used for the parameterization of the material field is known as Transfer learning FWI (TL-FWI). However, its application is largely restricted to uniform, rectilinear grids due to the architectural constraints of standard CNNs. In this work, we overcome these geometric limitations by proposing a novel Graph Convolutional Networks (GCNs) based TL-FWI. The physical domain is encoded as a graph allowing the presented approach to naturally extend TL-FWI to irregularly shaped geometries and complex boundary conditions that are common in industrial ultrasonic NDT applications. This also allows to employ the transfer learning strategy across dimensions where the GCN is pre-trained on a single rectangular 2D dataset as in [2] and employed for TL-FWI on various unseen 2D and 3D geometries described by the elastic wave equation. The results demonstrate that this ‘data-driven prior’ significantly accelerates convergence and improves reconstruction capabilities for FWI task using GCNs as compared to classical FWI. Numerical experiments also demonstrate the method’s robust generalization capabilities across diverse sensor configurations, source distributions, and material properties.