We derive a controlled generation objective within the framework of Variational Flow Matching (VFM), which casts flow matching as a variational inference problem. We demonstrate that controlled generation can be implemented two ways: (1) by way of end-to-end training of conditional generative models, or (2) as a Bayesian inference problem, enabling post hoc control of unconditional models without retraining. Furthermore, we establish the conditions required for equivariant generation and provide an equivariant formulation of VFM tailored for molecular generation, ensuring invariance to rotations, translations, and permutations. We evaluate our approach on both uncontrolled and controlled molecular generation, achieving state-of-the-art performance on uncontrolled generation and outperforming state-of-the-art models in controlled generation, both with end-to-end training and in the Bayesian inference setting. This work strengthens the connection between flow-based generative modeling and Bayesian inference, offering a scalable and principled framework for constraint-driven and symmetry-aware generation.

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.

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable $\mathrm{E}(n)$-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term *shared* simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

We present a formulation of flow matching as variational inference, which we refer to as variational flow matching (VFM). We use this formulation to develop CatFlow, a flow matching method for categorical data that is easy to implement, computationally efficient, and achieves strong results on graph generation tasks. In VFM, the objective is to approximate the posterior probability path, which is a distribution over possible end points of a trajectory. VFM admits both the original flow matching objective and the CatFlow objective as special cases. We also relate VFM to score-based models, in which the dynamics are stochastic rather than deterministic, and derive a bound on the model likelihood based on a reweighted VFM objective. We evaluate CatFlow on one abstract graph generation task and two molecular generation tasks. In all cases, CatFlow exceeds or matches performance of the current state-of-the-art models.

This paper presents $\mathrm{E}(n)$ Equivariant Message Passing Simplicial Networks (EMPSNs), a novel approach to learning on geometric graphs and point clouds that is equivariant to rotations, translations, and reflections. EMPSNs can learn high-dimensional simplex features in graphs (e.g. triangles), and use the increase of geometric information of higher-dimensional simplices in an $\mathrm{E}(n)$ equivariant fashion. EMPSNs simultaneously generalize $\mathrm{E}(n)$ Equivariant Graph Neural Networks to a topologically more elaborate counterpart and provide an approach for including geometric information in Message Passing Simplicial Networks, thereby serving as a proof of concept for combining geometric and topological information in graph learning. The results indicate that EMPSNs can leverage the benefits of both approaches, leading to a general increase in performance when compared to either method individually, being on par with state-of-the-art approaches for learning on geometric graphs. Moreover, the results suggest that incorporating geometric information serves as an effective measure against over-smoothing in message passing networks, especially when operating on high-dimensional simplicial structures.