## 1 Introduction
Although these methods reduce dependency on the initial model, they cannot guarantee accurate results when the initial model is of poor quality. In recent years, deep learning has gained traction in scientific and engineering fields, offering new solutions to complex geophysical problems, particularly in seismic inversion [14]. Traditional deep learning-based FWI is typically data-driven [15; 16], indicating that its effectiveness depends on the size and quality of the seismic dataset. Without constraints from geophysical forward modeling, traditional FWI results lack physical interpretability. To address this limitation, researchers have begun to integrate physics-informed neural networks (PINNs) with FWI to introduce physical constraints. PINNs, first proposed by [PERSON] et al. [17], solve nonlinear partial differential equations and have shown promise in seismology research. For example, [PERSON] et al. [18] proposed a mesh-free FWI method using PINNs for wave-equation solutions. [PERSON] et al. [19] developed a theory-guided system that incorporates wave equation physics into a network training loop usinga physics-driven data misfit as a loss function to enhance interpretability. Although PINN-based methods reduce the dependence of FWI on the initial model by introducing physical constraints, challenges remain in practical applications owing to limited prior information and data acquisition constraints. To address these challenges, [PERSON] et al. [20] proposed an implicit FWI (IFWI) using PINNs for seismic forward modeling. Their method employs a deep neural network (DNN) that emphasizes low-frequency learning, significantly reducing dependency on the initial model; however, its effectiveness depends substantially on the design of the deep learning model. Another approach involves leveraging neural networks to directly parameterize or generate the velocity model. For example, [PERSON] et al. [21] proposed a neural-network-based FWI (NNFWI) where a generative neural network is used to represent the velocity model. This method capitalizes on the neural network's ability to naturally introduce spatial correlations, acting as a regularization that helps suppress noise in the gradients and mitigate local minima, proving effective even for high-contrast structures and in the presence of noise. Furthermore, an alternative approach involves dynamic data matching, which enhances the correspondence between synthetic and observed data, thereby reducing dependency on the initial model. For instance, [PERSON] et al. [22] developed a full-waveform inversion technique centered on dynamic data matching of convolutional wavefields, employing one-dimensional Gaussian convolutional kernels to extract multi-scale features from individual time samples within seismic traces; these features are then used to identify optimally matched pairs between the observed and synthetic data. Utilizing first-arrival information to constrain FWI or improve the initial model is also an effective strategy. Some studies have focused on the joint inversion of travel time and waveform data [23; 24], leveraging their respective strengths to constrain the model. In recent years, deep learning has been applied to first-arrival picking to enhance picking efficiency and accuracy, indirectly aiding FWI [25; 26; 27]. While these methods alleviate the initial model problem to some extent, how to effectively utilize first-arrival information to dynamically guide the waveform inversion process and integrate it with PIRNNs remains a direction worthy of further investigation. In this study, we aim to reduce the impact of the initial model on FWI. Accordingly, we proposed a self-correlation progressive FWI method based on PIRNNs.