## 1 Introduction
Similarly, [PERSON] [10] and [PERSON] [11] proposed CNN-based one-dimensional deep-learning inversion strategies for electromagnetic and transient electromagnetic induction data. Furthermore, [PERSON] et al. [12] and [PERSON] et al. [13] applied CNNs and enhanced deep belief networks, respectively, to invert 2D MT data. These deep-learning-based techniques are entirely data-driven, relying on training datasets to establish mappings between inputs and outputs. Although these methods simplify the implementation of deep-learning-based inversion, they also present challenges related to interpretability and the significant costs of constructing large-scale datasets and performing the associated computations. Recent advancements have increasingly focused on physics-informed neural networks (PINNs) and the integration of prior knowledge into FWI as a response to longstanding challenges. Originally introduced by [PERSON] et al. (2019) [14] for solving complex physical problems, particularly nonlinear partial differential equations, PINNs combine deep learning with traditional scientific computing, offering an innovative framework for addressing both forward and inverse problems. Since their inception, the applications of PINNs have rapidly expanded to various domains. For instance, they have been employed in fluid dynamics for solving incompressible Navier-Stokes Equations (2021) [15] and addressing high-speed flow problems [16]. In seismology, PINNs have gained prominence through applications such as the Earthquake Transformer model developed by [PERSON] et al., which performs earthquake detection and phase identification simultaneously (2020) [17]. Additionally, PINNs have demonstrated considerable potential in solving wave Equations (2020) [18] and ray tracing problems (2021) [19], underscoring their versatility across a broad spectrum of scientific and engineering challenges. Within the FWI domain, [PERSON] et al. [20] utilized PINNs by incorporating seismic data and prior initial models as inputs for network training, effectively reducing the dependence on extensive datasets. Similarly, [PERSON] et al. [21] proposed a novel method leveraging PINNs to solve FWI problems. This approach employs neural networks to solve the partial differential equations governing acoustic wave propagation, utilizing their mesh-free properties to accommodate diverse boundary conditions. [PERSON] et al. (2020) [22] further extended PINN applications in FWI by developing a theory-guided system that integrates the forward modeling physics of wave equations directly into the training loop. By employing physics-based data mismatch as the loss function, this system achieves unsupervised deep learning, reframing the process as a physics-guided FWI technique. Compared to purely data-driven methods, PINN-based approaches and FWI methods that incorporate prior knowledge exhibit greater robustness against noise. Furthermore, unlike traditional FWI techniques, PINN-based methods offer a flexible framework for integrating diverse data types and structural prior knowledge, thereby enhancing inversion accuracy and reliability. Their versatility across various wave propagation models makes them particularly advantageous for addressing complex geophysical problems. However, the computational demands of PINN-based methods, especially for large-scale three-dimensional applications, remain a significant barrier to their practical implementation. To address the challenge of high computational resource consumption, this study introduces a dual-branch physics-informed recurrent neural network vertical seismic profiling (PIRNN VSP) FWI architecture. The central concept involves constructing multiscale branches and facilitating information transfer across these scales. Specifically, the proposed method enables the propagation of information from the original-scale branch to a downscaled branch, effectively performing full-waveform inversion while significantly reducing GPU resource consumption during the inversion process.