The field of inverse problems and data synthesis is witnessing significant developments, driven by advances in machine learning, signal processing, and statistical methods. One of the key directions is the integration of deep learning-based models with physical constraints to improve the accuracy and robustness of inverse problem solutions. This includes the use of projection-based correction methods to ensure consistency with the forward model, as well as the development of novel architectures that can handle complex, high-dimensional data distributions. Another important area of research is the application of diffusion models to inverse problems, which has shown promise in capturing complex data distributions and providing efficient sampling mechanisms. Furthermore, there is a growing interest in using score-based methods to explore the geometry of data manifolds, which can lead to new insights into the structure of complex data sets. Noteworthy papers in this area include: mmMirror, which proposes a novel system for precise, device-free NLoS localization using a Van Atta Array-based millimeter-wave reconfigurable intelligent reflecting surface. What's Inside Your Diffusion Model, which introduces a score-based Riemannian metric to explore the intrinsic geometry of the data manifold learned by diffusion models. Guided Diffusion Sampling on Function Spaces, which proposes a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements.