While denoising diffusion and flow matching have driven major advances in generative modeling, their application to tabular data remains limited, despite its ubiquity in real-world applications. To this end, we develop *TabbyFlow*, a variational Flow Matching (VFM) method for tabular data generation. To apply VFM to data with mixed continuous and discrete features, we introduce **Exponential Family Variational Flow Matching (EF-VFM)**, which represents heterogeneous data types using a general exponential family distribution. We hereby obtain an efficient, data-driven objective based on moment matching, enabling principled learning of probability paths over mixed continuous and discrete variables. We also establish a connection between variational flow matching and generalized flow matching objectives based on Bregman divergences. Evaluation on tabular data benchmarks demonstrates state-of-the-art performance compared to baselines.

Natural gradient methods significantly accelerate the training of Physics-Informed Neural Networks (PINNs), but are often prohibitively costly. We introduce a suite of techniques to improve the accuracy and efficiency of energy natural gradient descent (ENGD) for PINNs. First, we leverage the Woodbury formula to dramatically reduce the computational complexity of ENGD. Second, we adapt the Subsampled Projected-Increment Natural Gradient Descent algorithm from the variational Monte Carlo literature to accelerate the convergence. Third, we explore the use of randomized algorithms to further reduce the computational cost in the case of large batch sizes. We find that randomization accelerates progress in the early stages of training for low-dimensional problems, and we identify key barriers to attaining acceleration in other scenarios. Our numerical experiments demonstrate that our methods outperform previous approaches, achieving the same $L^2$ error as the original ENGD up to $75\times$ faster.