# Deep Learning Based Inversion of Locally Anisotropic Weld Properties from Ultrasonic Array Data
## 4 Discussion  
There are several limitations to the proposed framework that are yet to be overcome in the current work. Firstly, the experimental geometries considered here are confined to those where a transducer array is placed on top of the weld. This brings some practical challenges due to the non-planar topography of the weld cap. While these challenges could be overcome with the use of remote sensing systems (for example laser ultrasonics [43]), future studies will examine DNNs trained on alternative experimental geometries which can be applied in situ, for example where the arrays are placed away from the weld cap and mounted on wedges. In this paper, we have also used a simple background model (the Ogilvy model [29]) to constrain the inversion, by generating training data that consists only of random perturbations to this background model. The Ogilvy model is a relatively basic description of how grain orientations are distributed through welds, and it may  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Defect A** & **Defect B** & **Defect C** \\ \hline
**TFM** & 11.7 dB & 14.1 dB & 10.5 dB \\
**TFM+** & 14.3 dB & 14.1 dB & 15.8 dB \\ \hline \hline \end{tabular}
\end{table}
Table 2: Signal-to-noise ratio (SNR) for TFM and TFM+ flaw images shown in Figure 9.  
Figure 10: The difference between a travel time map that assumes the constant wave speed of the parent material throughout the domain and one that accounts for the predicted grain map, using centrally placed transducer element.  
be that more advanced weld models, such as the MINA model [13] which takes in more information about welding parameters, can be used as an alternative model to constrain the DNN inversion. Another possible avenue for improvement lies in further development of the model used to train the network. In its current state, although the AMSFMM is fast and provides travel time estimates within a desired tolerance for heterogeneous and locally anisotropic structures, it suffers from some biasing toward the underlying finite difference stencils, causing numerical errors to propagate in large homogeneous regions. Consideration of the anisotropic Eikonal equation and the curvature of the wavefront could significantly diminish these errors and provide more accurate models of the travel times, which could in turn reduce the requirement for the data calibration step as discussed in Section 2.4. Finally, the current approach provides a single approximation of the weld map and so, in real-world applications where the underlying weld geometry is unknown, it is not possible to definitively measure or comment on the accuracy of the predicted grain orientations. Estimating the uncertainty of the predicted grain orientations through the use of more sophisticated networks such as mixture density networks [25] or Bayesian neural networks [44] thus presents an important future step to improve the current framework.  
The next step for testing this framework is application to data acquired from a controlled physical experiment, where ground truth of the weld structure is made available through destructive methods such as the electron backscatter diffraction technique. However, several assumptions need to be relaxed to reach the point where the presented DNN framework can be reliably applied to experimentally acquired data. Although it has been tested using Onscale data (which has undergone some experimental validation [32]), the current approach makes no allowance for background noise in the acquired data and uncertainty in the transducer locations. To overcome these assumptions, additional noise can be incorporated into the modelled travel time data representing the expected level of background noise or the uncertainty in the transducer locations.