## 1 Introduction
Notable deep generative models include the pixel recurrent neural network (PixelRNN) and pixel convolutional neural network (PixelCNN) [14; 15], the variational auto-encoder (VAE) [16], and generative adversarial networks (GANs) [17]. Among these, GANs stand out for their adversarial learning framework, which can overcome the limitations of traditional stochastic simulation methods such as MPS to achieve automatic construction. First, GANs, with their adversarial learning framework, can effectively capture the distributional characteristics of the training datasets, overcoming the dependence of MPS on predefined templates. Additionally, MPSis constrained by local multi-point statistical information, making it difficult to capture complex non-stationary spatial structures within geological bodies. By contrast, GANs leverage deep network layers to fit complex features, enabling more realistic and diverse reservoir modeling. Moreover, GANs allow for efficient inferences once trained, generating complex structures in a short time without extensive recalculations. This ensures not only the diversity of the simulation results but also significantly improves simulation efficiency. For example, [PERSON] et al. [18] employed GANs to implicitly encode and replicate the distribution of a dataset, achieving the rapid and precise reconstruction of porous media across multiple scales. [PERSON] et al. [19] investigated a GAN-based reservoir modeling method to capture facies patterns directly from TIs without conditional data. In a different context, for the multispectral satellite image set, [PERSON] and [PERSON] [20] implemented a stochastic simulation of 2D river facies with unconditional constraints. [PERSON] et al. [21] proposed a concurrent multi-stage GAN based on a single TI, which adopts a pyramid structure for multi-scale representation of spatial patterns and can capture the feature information corresponding to TIs at different scales. Furthermore, [PERSON] et al. [22] developed an image-to-image transformation technique with different dimensions based on the bijective consistency between the potential vectors and the outputs and coupled multiple networks to achieve a fine representation. Thus, it is evident that GANs are the superior choice for the stochastic simulation of reservoirs, rather than merely replicating the geological patterns in TIs. By adversarially learning the underlying distribution characteristics of TIs, GANs ensure both the accuracy and diversity of simulation results. However, the inherent complexity of geological structures, characterized by their highly nonlinear, nonstationary, and anisotropic heterogeneous properties, makes it challenging for GANs to reproduce the data distribution features. In particular, for nonstationary spatial attribute fields, the GANs based on deep convolution have a fixed-size receptive field, which ignores the statistical feature variations in different scale ranges during the feature extraction process, resulting in inadequate fitting of the latent distribution of the training dataset. Although SinGAN achieves multi-scale feature extraction through independently trained GAN networks, this can lead to instability across layers. Moreover, as an unconditional generative model, it requires additional strategies for conditional simulation, reducing its flexibility in conditional modeling tasks [23]. Moreover, since training GANs requires frequent hyperparameter adjustment and network redesign, the generator may fall into local optima and increase the risk of mode collapse. Especially in the early stage of training, when the generated distribution significantly deviates from the real, the loss metrics based on KL divergence or JS divergence are prone to gradient explosion [24; 25]. In addition, due to the limitations of the network structure, it is challenging to effectively incorporate conditioning data into the reconstruction process. Sparsely distributed conditioning data may be misinterpreted as noise, thus failing to influence global spatial features [26; 27].