Abstract: In this talk, we introduce the Signatured Deep Fictitious Play algorithm for solving Mean-Field Games (MFGs) with common noise. Existing deep learning methods fix the sampling common noise paths and then solve the corresponding MFGs. This leads to a nested-loop structure with millions of simulations of common noise paths in order to produce accurate solutions, which results in prohibitive computational cost and limits the applications to a large extent. Based on signatures from the rough path theory, we propose a novel single-loop algorithm, by which we can work with the unfixed common noise setup to avoid the nested-loop structure and reduce the computational complexity significantly. We showcase its efficiency across various applications, including linear-quadratic MFGs and mean-field portfolio games (including extended MFGs). Expanding on the application of signatures, we introduce Directed Chain Generative Adversarial Networks (DC-GANs) for multimodal time series data generation. To construct the generator, we embed time series datasets into drifts and diffusions of directed chain SDEs; and we employ the signature in the discriminator. Our experiments show that DC-GANs consistently outperform benchmarks across diverse datasets, marking a significant advancement in multimodal time series generation.