## 3 Methodology
### Network Structure  
The network architecture of the dual-PIRNN is illustrated in Figure 2. The proposed inversion process is divided into two branches. One branch performs forward modeling on the original-scale model to compute the loss at the original-scale. The second branch, referred to as the downscaled branch, conducts FWI by receiving the scale-decomposed loss from the original-scale model. The velocity parameters are updated on the downscaled model, which is subsequently upsampled and fed back into the original-scale branch for further forward modeling, repeating this cycle. The key distinction between the downscaled and original-scale models lies in their parameter settings. For instance, parameters such as the sampling rate \(dt\) of the source function, the number of time points \(NT\) for numerical simulation, and the number of grid points \(nz\) and \(nx\) exhibit a multiplicative relationship between the two scales. During each iteration of the inversion process, forward modeling is first performed on the initial original-scale model using the SGFD RNN operator to obtain the simulated shot-gather data \(d_{cal}\). The loss at the original scale, \(Loss_{original}\), is then calculated using the L1 norm between \(d_{obs}\) and \(d_{cal}\), as expressed in (9):  
\[Loss_{original}=d_{obs}-d_{cal}. \tag{9}\]  
Here, the \(d_{cal}\) and \(d_{obs}\) obtained through the VSP observation system are time-series matrices with a size of \(NT\times nz\). Consequently, the calculated \(Loss_{original}\) is also a time-series matrix with a size of \(NT\times nz\). a downsampling method based on average pooling is then applied to extract \(Loss_{original}\), yielding \(Loss_{down}\) that matches the required data size for the downscaled branch model. Forward modeling is subsequently performed using the SGFD RNN operator on the downscaled branch to generate the VSP shot-gather \(d_{Dcal}\) on  
Figure 2: Dual-PIRNN process.  
the downscaled model. By summing \(Loss_{down}\) and \(d_{Dcal}\), the observed record \(d_{Dobs}\) on the downscaled model is obtained, as expressed in (10):  
\[\begin{split} Loss_{down}=ExtractionOperator(Loss_{original})\\ d_{Dobs}=dDcal+Loss_{down}\end{split}. \tag{10}\]  
The extraction operator refers to a downsampling operator based on average pooling. Once the downscaled observed data, \(d_{Dobs}\) is obtained, the inversion process--detailed in Section 2 and based on the PIRNN--can be executed on the downscaled model to update its velocity parameters. The mean squared error (MSE) is employed as the loss function, and the Adam optimizer is used for parameter updates. The loss function for the downscaled model is presented in (11):  
\[Loss=MSE(d_{Dobs},d_{Dcal}). \tag{11}\]  
The calculated \(Loss\) is propagated backward to update the velocity parameter \(v\) in the downscaled model. Once updated, the velocity parameters \(v\) are upscaled to revert to the original scale, preparing them for the next full-scale forward modeling iteration. This sequence represents the inversion flow within each cycle of the dual-PIRNN network structure (Figure 2). During this process, inversion occurs on the downscaled model, Compared to traditional FWI, which performs inversion at the original scale, the dual-branch architecture of the PIRNN further minimizes GPU resource consumption by leveraging scale decomposition: the original-scale shot gathers are transformed into downscaled models, which require fewer grid points and shorter forward modeling time. Since the PIRNN-based full-waveform inversion records gradient information for the entire spatiotemporal field, its resource consumption is directly proportional to the forward modeling time and the number of grid points. By reducing both factors through downscaling, the dual-branch architecture achieves significant savings in GPU memory and computational costs. Consequently, computational costs are directly reduced.