The Bayesian inference approach is widely used to tackle inverse problems due to its versatile and natural ability to handle ill-posedness. However, it often faces challenges when dealing with situations involving continuous fields or large-resolution discrete representations (high-dimensional). Moreover, the prior distribution of unknown parameters is commonly difficult to be determined. In this study, an Operator Learning-based Generative Adversarial Network (OL-GAN) is proposed and integrated into the Bayesian inference framework to handle these issues. Unlike most Bayesian approaches, the distinctive characteristic of the proposed method is to learn the joint distribution of parameters and responses. By leveraging the trained generative model, the posteriors of the unknown parameters can theoretically be approximated by any sampling algorithm (e.g., Markov Chain Monte Carlo, MCMC) in a low-dimensional latent space shared by the components of the joint distribution. The latent space is typically a simple and easy-to-sample distribution (e.g., Gaussian, uniform), which significantly reduces the computational cost associated with the Bayesian inference while avoiding prior selection concerns. Furthermore, incorporating operator learning enables resolution-independent in the generator. Predictions can be obtained at desired coordinates, and inversions can be performed even if the observation data are misaligned with the training data. Finally, the effectiveness of the proposed method is validated through several numerical experiments.

翻译：贝叶斯推断方法因其灵活且自然地处理不适定问题的能力而被广泛应用于逆问题求解。然而，当涉及连续场或高分辨率离散表示（高维）的情况时，该方法常面临挑战。此外，未知参数的先验分布通常难以确定。本研究提出一种基于算子学习的生成对抗网络（OL-GAN），并将其集成到贝叶斯推断框架中以应对上述问题。与大多数贝叶斯方法不同，所提方法的显著特征在于学习参数与响应的联合分布。通过利用训练后的生成模型，原则上可在联合分布各分量共享的低维潜空间中，采用任意采样算法（例如马尔可夫链蒙特卡洛MCMC）来近似未知参数的后验分布。该潜空间通常为易于采样的简单分布（如高斯分布、均匀分布），从而显著降低贝叶斯推断的计算成本，同时避免先验选择问题。此外，引入算子学习使生成器实现分辨率无关性。即便观测数据与训练数据存在不对准，亦可在目标坐标处获得预测结果并进行反演计算。最后，通过多个数值实验验证了所提方法的有效性。