## 1 Introduction  
Full-waveform inversion (FWI) is a pivotal technique in geophysical exploration, extensively utilized to derive detailed subsurface velocity structures. These structural insights are integral to applications such as resource exploration and seismic hazard analysis. Traditional velocity-inversion methods, including travel-time tomography and ray tracing, rely on limited data, focusing primarily on first-arrival times or specific ray paths. In contrast, FWI leverages the full-waveform information from seismic records, enabling a more comprehensive estimation of subsurface medium parameters. By utilizing the entire seismic wavefield, FWI achieves superior accuracy and detail in reconstructing the Earth's subsurface properties, representing a significant advancement in characterizing complex geological structures. The concept of FWI was first introduced in the 1980s by [PERSON] and [PERSON] [1]. It uses wavefield information contained in seismic datasets to reconstruct subsurface parameters by minimizing the discrepancy between recorded observational data (real seismic records) and predicted data (synthetic records derived from an initial model). As a highly nonlinear method, FWI incurs substantial computational costs. [PERSON] et al. (1998) [2] implemented FWI in the frequency domain, improving computational efficiency; however, its application is constrained by the need for low-frequency data acquisition. [PERSON] and [PERSON] (2008) [3] extended FWI to the Laplace and Laplace-Fourier domains, enhancing imaging capabilities for deep structures but further increasing computational demands. A recurring challenge in FWI is cycle-skipping, which arises when the initial model significantly deviates from the actual subsurface model. To address this, [PERSON] and [PERSON] (2016) [4] introduced adaptive waveform inversion (AWI), employing a traceable objective function to mitigate cycle-skipping and reduce dependence on low-frequency data, albeit with potential compromises in resolution. To account for the complexity of subsurface media, [PERSON] et al. (2018) [5] enhanced FWI by simultaneously inverting multiple physical parameters (e.g., velocity-density, modulus-density, impedance-density), thereby improving the comprehensiveness of subsurface characterization. However, this approach increased the problem's complexity and ill-posedness, necessitating additional prior information and computational resources. Further refinements were made by [PERSON], [PERSON] et al. (2019) [6], who defined a new objective function using optimal transport theory, enhancing the utilization of low-frequency information. Despite these advancements, FWI continues to face challenges in balancing computational efficiency, accuracy, and reliance on initial models. In recent years, significant progress in large-scale computing and the rapid development of deep-learning technologies have revolutionized various computational engineering fields. Deep learning has shown remarkable success in structural health monitoring (SHM). For instance, [PERSON] et al. [7] developed a 1D CNN model for concrete stress monitoring using smart aggregate-based impedance signals, demonstrating high accuracy in stress estimation even with noisy data. Similarly, [PERSON] et al. [8] proposed an automated bolt-loosening monitoring method by integrating impedance technique with deep learning, achieving precise identification of bolt locations and loosening degrees with minimal pre-processing requirements. In the field of geophysical exploration, deep-learning techniques have also demonstrated promising applications. Within the FWI domain, traditional deep-learning techniques generally involve three key steps: generating training data, constructing and training deep neural networks, and employing the network for prediction and subsurface structure imaging. For instance, [PERSON] et al. [9] utilized convolutional neural networks (CNNs) for seismic impedance inversion.