This proposal seeks computational resources to advance foundational and practical aspects of generative AI and sequence modeling, using tools and techniques from applied probability, dynamical systems, and modern machine learning. Our research aims to understand, improve, and scale generative models—particularly diffusion models and their generalizations such as flow matching—while addressing critical challenges in scientific and engineering applications. Generative models learn complex data distributions and are central to recent advances in AI. Among these, diffusion models have shown remarkable success across modalities like images, audio, and text by learning to reverse a stochastic noising process. However, they suffer from high computational cost and are typically restricted to Gaussian assumptions and specific score-based objectives. Flow matching offers a more flexible alternative by framing generation as learning deterministic or stochastic flows between distributions. It generalizes diffusion models and can enable faster sampling with greater modeling adaptability. Our recent work introduces a novel framework for applying flow matching to spatio-temporal data, targeting probabilistic forecasting in dynamical systems. We demonstrate that model performance is highly sensitive to the choice of the probability path and propose a new class of probability paths tailored to time series. We plan to systematically investigate the impact of design choices in flow-based generative models, including the selection of loss parametrizations, time-sampling schemes, and covariance structures of latent processes. These insights are critical for improving convergence, sample quality, and computational efficiency. Additionally, we aim to extend flow matching to higher-order diffusion models that incorporate auxiliary momentum or memory variables. These formulations promise more stable and efficient training while reducing discretization error and accelerating sampling. The proposed research will combine theoretical analysis with empirical validation across standard and scientific datasets. A key focus is on robustness: how generative models respond to data perturbations and how to enhance their resilience, particularly in domains where data is noisy, sparse, or irregularly sampled. Inspired by recent work on noise-based regularization and stability training, we will develop new techniques for robust training of flow matching models and evaluate their effectiveness on downstream tasks like conditional generation and forecasting. Scientific applications, such as weather prediction, provide an ideal testbed for our methods due to their inherent complexity and real-world importance. We will use simulated data from known dynamical systems and real-world benchmarks to evaluate our models’ generalization and robustness. Where appropriate, we aim to collaborate with industrial partners on practical applications of generative models to sequential data. Overall, this project combines foundational contributions with practical advances in generative AI. GPU access is critical to support the training of state-of-the-art models and conduct extensive empirical studies. The computational resources will enable us to test theoretical insights at scale, bridging the gap between theory and application in modern generative modeling.