## 4 Numerical Examples
### GPU Memory Usage Analysis  
Having validated the effectiveness of our proposed dual-PIRNN FWI network architecture, we now address GPU resource consumption issues during the inversion process. In the deep-learning framework, GPU memory usage can be primarily categorized into four components: model parameters, gradients of model parameters, optimizer states, and intermediate activation values. In the proposed dual-PIRNN FWI framework, the model parameters are defined as the velocity parameters. The gradients of the model parameters refer to the gradients of the loss function with respect to the velocity parameters, which are automatically calculated and stored during the backpropagation process. For the optimizer states, since the Adam optimizer is employed, the first-order and second-order moment states for each model parameter must be maintained. This results in memory consumption approximately twice that of the model parameters. Lastly, in the context of PIRNN-based FWI, the intermediate activation values correspond to the wavefield information within all RNN units across the entire spatiotemporal field. Wavefield data for each time point are sequentially calculated and stored according to the time-series relationship, consuming a significant amount of GPU memory. From a resource utilization perspective, decomposing both time and spatial scales reduces the number of time points and spatial grid points required for numerical simulation. This decomposition significantly decreases GPU memory consumption for intermediate activation values, thereby improving resource efficiency during the inversion process. Compared to traditional FWI methods that require storing the complete forward wavefield for adjoint-state calculations, our dual-PIRNN approach demonstrates distinct advantages in memory management. While conventional methods need to store wavefields at all time steps for backpropagation, leading to memory requirements scaling with \(O(NT)\) (where N is the spatial grid size and T is the number of time steps), our method's memory usage primarily comes from storing gradients of intermediate activations. As demonstrated in our previous simulation analysis  
Figure 17: Inversion results of Marmousi model with initial model at \(\sigma\) = 20. (**a**) the inversion with DownTimeScale. (**b**) the inversion with DownTimeScale. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Model** & **Scale Strategy** & **R2** & **SSIM** & **NCC** \\ \hline \multirow{2}{*}{Homogeneous} & DownTime & 0.9574 & 0.5478 & 0.9805 \\  & DownTimeSpace & 0.7978 & 0.6254 & 0.9449 \\ \hline \multirow{2}{*}{Marmousi (\(\sigma\) = 5)} & DownTime & 0.9254 & 0.9751 & 0.9876 \\  & DownTimeSpace & 0.8927 & 0.7212 & 0.9477 \\ \hline \multirow{2}{*}{Marmousi (\(\sigma\) = 10)} & DownTime & 0.9421 & 0.8746 & 0.9711 \\  & DownTimeSpace & 0.8736 & 0.6972 & 0.9373 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Quantitative metrics for different inversion scenarios. (Figures 13-16), this approach maintains comparable inversion accuracy while offering flexible memory-accuracy trade-offs through scale decomposition. In our experiments, we applied three scales to the Marmousi velocity model: original scale, downtimescale, and downtimespacescale. For the original scale, the dual-PIRNN FWI network architecture is not required; instead, inversion is directly performed following the PIRNN-based inversion process, utilizing 1600 time points. For inversion under downtimescale, based on scale decomposition, the required number of time points for numerical simulation is reduced to 800, while the grid size remains unchanged. Theoretically, this reduces the storage needed for wavefield information in the intermediate activation values by half. However, to perform an inversion on the reduced-scale using the dual-PIRNN network structure, forward modeling is first conducted on the original-scale branch to obtain shot-gather data, which is then transformed to the reduced -scale branch according to the scale decomposition.