Normalizing Flows (NFs) are powerful and efficient models for density estimation. When modeling densities on manifolds, NFs can be generalized to injective flows but the Jacobian determinant becomes computationally prohibitive. Current approaches either consider bounds on the log-likelihood or rely on some approximations of the Jacobian determinant. In contrast, we propose injective flows for star-like manifolds and show that for such manifolds we can compute the Jacobian determinant exactly and efficiently. This aspect is particularly relevant for variational inference settings, where no samples are available and only some unnormalized target is known. Among many, we showcase the relevance of modeling densities on star-like manifolds in two settings. Firstly, we introduce a novel Objective Bayesian approach for penalized likelihood models by interpreting level-sets of the penalty as star-like manifolds. Secondly, we consider probabilistic mixing models and introduce a general method for variational inference by defining the posterior of mixture weights on the probability simplex.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differential form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.