A Lagrangian View of Flow Matching

Peyman Milanfar

Home/Archive/arXiv

arXiv · 2609.00198

A Lagrangian View of Flow Matching

Peyman Milanfar

View original source ↗Read source document ↗Export metadata ↓

Abstract

Modern explicit-time generative models, such as Flow Matching [Lipman et al., 2023] and Rectified Flow [Liu et al., 2023], are typically derived top-down via Optimal Transport and the continuity equation. This standard Eulerian approach focuses on the macroscopic transport of probability mass. In this paper, we present an alternative, bottom-up mechanical derivation grounded in a Lagrangian (particle-centric) perspective. By analyzing the local Taylor expansion of a continuous denoiser, we motivate a strict invariance condition required for optimal, singlestep generation: the conservation of target identity. Enforcing this condition yields a governing quasi-linear advection Partial Differential Equation (PDE). We demonstrate that solving this PDE via the Method of Characteristics analytically yields the straight-line trajectories of Flow Matching. This geometric perspective isolates the Jacobian of the denoiser as the primary source of trajectory curvature, providing a direct mathematical explanation for why straight-line flows enable massive step sizes, and why empirical models require distillation to flatten intersecting characteristics.

Explore related subjects

cs.CV physics.flu-dyn

Keep this discovery

BibTeX RIS

Peyman Milanfar. 2026-09-05. A Lagrangian View of Flow Matching. https://arxiv.org/abs/2609.00198

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.CV subject physics.flu-dyn subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

physics.flu-dyn↗

arXiv2026

Geometry Parameterisation via a Structural Modal Basis for Aerodynamic Shape Optimisation

Aircraft outer mould lines are often heavily constrained early by payload requirements, manufacturability limits, and low-observability considerations. The remaining aerodynamic design space is typically small, highly constrained, and difficult to represent using a concise and physically meaningful set of design variables. A structural modal parameterisation method (MPM) is introduced for such problems. The method constructs a tunable pseudo-structure and solves its eigenvalue problem to obtain a modal basis, from which a selected subset of modes parameterises the geometry. Boundary conditions, stiffness distribution, density, thickness, and added masses are treated as intentional basis-design variables that shape the admissible deformation space prior to optimisation. The resulting design coordinates are independent of any specific flow solver, making the approach broadly applicable. For tightly constrained aerodynamic shape optimisation problems, it provides a practical balance of parameter conciseness, robust constraint awareness, and geometric flexibility.

math.NA↗

arXiv2026

Physics-Informed Neural Networks for Depth-Averaged Granular Avalanche Dynamics on Curved Topography

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving governing equations, but their application to granular avalanche dynamics over curved terrain remains largely unexplored. This study extends a depth-averaged PINN formulation based on the Savage-Hutter equations to an exponentially curved chute with spatially varying inclination and a strain-rate-dependent Mohr-Coulomb earth-pressure closure. The model is validated against measured front- and rear-edge trajectories from a laboratory granular-avalanche experiment, with selected observations withheld from training. A staged temporal curriculum proved essential for accurate prediction, reducing the held-out trajectory error by approximately two orders of magnitude compared with training over the full time domain from the outset. Sparse-data experiments further showed that observation placement was more influential than observation number within the configurations tested. Four observations bracketing the transition from acceleration to deceleration achieved nearly the same accuracy as the eight-observation reference configuration, whereas observations clustered at early or late times performed poorly. The results demonstrate the importance of both training strategy and informative data placement when applying PINNs to granular flows over curved topography.

cond-mat.soft↗