## 7 Conclusions  
This study presents a novel Physics-Informed Recurrent Neural Network Full Wave-form Inversion method with first-arrival constraints (PIRNN-FAFWI) to reduce the sensitivity of FWI to initial velocity models. Our method utilized the information contained in first-arrival waveforms to select seismic traces dynamically for loss calculations and effectively reduced the impact of inaccurate initial models on the inversion process. This dynamic selection, combined with a progressive strategy that gradually introduced more seismic traces as the inversion progressed, ensured a stable and robust inversion workflow. Our method integrated both first-arrival and reflected wave data, by computing the cross-correlation between the observed and simulated data, we extracted the first arrival time differences and used them to select appropriate seismic traces for inversion, thereby improving the robustness and resolution of the results. Through numerical experiments using the 2D acoustic Marmousi model and 2007 BP TTI anisotropic velocity benchmark model, we demonstrated the effectiveness and robustness of this method in reducing the dependence on initial velocity models. The FAFWI method inverted accurate velocity models, even with poor initial models. Quantitative analyses using metrics such as R\({}^{2}\), SSIM, and NCC further confirmed the excellent performance of the proposed method in terms of linear correlation, structural similarity, and overall similarity with the true velocity model. In future studies, we will employ more precise first-arrival picking methods and investigate noise suppression techniques to further enhance the FAFWI performance.  
In future studies, we will employ more precise first-arrival picking methods and investigate noise suppression techniques to further enhance the FAFWI-PIRNN performance. Furthermore, to more comprehensively assess the robustness and generalization capabilities of the proposed method, future work could explore its performance under a wider and more diverse range of initial velocity models, encompassing varying structural complexities, noise levels, and degrees of deviation from the true model. Additionally, although the first-arrival constraints in FAFWI-PIRNN effectively address cycle-skippingrelated local minima, further investigation into the optimization process itself, such as the impact of different optimizers, learning rate schedules, and regularization techniques on potential oscillations or suboptimal local minima encountered during the inversion, could lead to further improvements in solution quality and convergence reliability.  
Conceptualization, C.L. and J.L. ([PERSON]); methodology, J.L. ([PERSON]); software, J.L. ([PERSON]); validation, J.L. ([PERSON]) and H.C.; formal analysis, L.Q.; investigation, L.Q.; resources, J.G.; data curation, J.L. ([PERSON]) and J.G.; writing--original draft preparation, J.L. ([PERSON]); writing--review and editing, J.L. ([PERSON]); visualization, C.L. and J.L. ([PERSON]); supervision, C.L.; project administration, C.L. and J.L. ([PERSON]); funding acquisition, C.L. and H.C. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (Grant No. 42474168).  
Not applicable.  
The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.  
We are grateful to the reviewers for their valuable comments.  
The authors declare no conflicts of interest.