# Flow Matching Ergodic Coverage arXiv: https://arxiv.org/abs/2504.17872 ยท HTML: https://mlsanity.com/paper/2504.17872/ ## Key facts - Cited by 3 papers in the map (data through 14 Oct 2026). - Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models. - Builds on 31 of its 61 references that are in the map. - First posted on arXiv 24 Apr 2025. ## Placed in - Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/) ## Builds on (most cited first, 30 of 31) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016) - [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019) - [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019) - [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017) - [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023) - [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017) - [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019) - [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024) - [Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matching](https://mlsanity.com/paper/2409.07343/) (2024) - [Message Passing Stein Variational Gradient Descent](https://mlsanity.com/paper/1711.04425/) (2017) - [Efficient Trajectory Forecasting and Generation with Conditional Flow Matching](https://mlsanity.com/paper/2403.10809/) (2024) - [Ergodic Exploration of Distributed Information](https://mlsanity.com/paper/1708.09352/) (2017) - [Ergodic imitation: Learning from what to do and what not to do](https://mlsanity.com/paper/2103.17098/) (2021) - [Real-Time Area Coverage and Target Localization using Receding-Horizon Ergodic Exploration](https://mlsanity.com/paper/1708.08416/) (2017) - [Ergodic Exploration using Tensor Train: Applications in Insertion Tasks](https://mlsanity.com/paper/2101.04428/) (2021) - [Fast Ergodic Search with Kernel Functions](https://mlsanity.com/paper/2403.01536/) (2024) - [Decentralized Ergodic Control: Distribution-Driven Sensing and Exploration for Multi-Agent Systems](https://mlsanity.com/paper/1806.05220/) (2018) - [Ergodic Specifications for Flexible Swarm Control: From User Commands to Persistent Adaptation](https://mlsanity.com/paper/2006.06081/) (2020) - [Model predictive altitude and velocity control in ergodic potential field directed multi-UAV search](https://mlsanity.com/paper/2401.02899/) (2024) - [Stein Variational Ergodic Search](https://mlsanity.com/paper/2406.11767/) (2024) - [Embodied Active Learning of Generative Sensor-Object Models](https://mlsanity.com/paper/2410.11130/) (2024) - [Ergodic Exploration using Binary Sensing for Non-Parametric Shape Estimation](https://mlsanity.com/paper/1709.01560/) (2017) - [Task-Based Hybrid Shared Control for Training Through Forceful Interaction](https://mlsanity.com/paper/1911.07983/) (2019) - [Ergodic Trajectory Optimization on Generalized Domains Using Maximum Mean Discrepancy](https://mlsanity.com/paper/2410.10599/) (2024) - [Data-Driven Measurement Models for Active Localization in Sparse Environments](https://mlsanity.com/paper/1806.00112/) (2018) ## Built on by (most cited first, 3 of 3) - [Volumetric Ergodic Control](https://mlsanity.com/paper/2511.11533/) (2025) - [Crazyflow: An Accurate, GPU-Accelerated, Differentiable Drone Simulator in JAX](https://mlsanity.com/paper/2606.01478/) (2026) - [Ergodic Trajectory Design by Learned Pushforward Maps: Provable Coverage via Conditional Flow Matching](https://mlsanity.com/paper/2605.13063/) (2026) ## Abstract (arXiv) Ergodic coverage effectively generates exploratory behaviors for embodied agents by aligning the spatial distribution of the agent's trajectory with a target distribution, where the difference between these two distributions is measured by the ergodic metric. However, existing ergodic coverage methods are constrained by the limited set of ergodic metrics available for control synthesis, fundamentally limiting their performance. In this work, we propose an alternative approach to ergodic coverage based on flow matching, a technique widely used in generative inference for efficient and scalable sampling. We formally derive the flow matching problem for ergodic coverage and show that it is equivalent to a linear quadratic regulator problem with a closed-form solution. Our formulation enables alternative ergodic metrics from generative inference that overcome the limitations of existing ones. These metrics were previously infeasible for control synthesis but can now be supported with no computational overhead. Specifically, flow matching with the Stein variational gradient flow enables control synthesis directly over the score function of the target distribution, improving robustness to the unnormalized distributions; on the other hand, flow matching with the Sinkhorn divergence flow enables an optimal transport-based ergodic metric, improving coverage performance on non-smooth distributions with irregular supports. We validate the improved performance and competitive computational efficiency of our method through comprehensive numerical benchmarks and across different nonlinear dynamics. We further demonstrate the practicality of our method through a series of drawing and erasing tasks on a Franka robot.