# NSFlow: End-to-End Differentiable Neuro-Symbolic Optical Flow for Visual Odometry

arXiv: https://arxiv.org/abs/2609.06074 · HTML: https://mlsanity.com/paper/2609.06074/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 57 references that are in the map.
- First posted on arXiv 5 Sep 2026.

## Builds on (most cited first, 30 of 34)

- [RAFT: Recurrent All-Pairs Field Transforms for Optical Flow](https://mlsanity.com/paper/2003.12039/) (2020)
- [SuperPoint: Self-Supervised Interest Point Detection and Description](https://mlsanity.com/paper/1712.07629/) (2017)
- [FlowNet: Learning Optical Flow with Convolutional Networks](https://mlsanity.com/paper/1504.06852/) (2015)
- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [ORB-SLAM3: An Accurate Open-Source Library for Visual, Visual-Inertial and Multi-Map SLAM](https://mlsanity.com/paper/2007.11898/) (2020)
- [FlowNet 2.0: Evolution of Optical Flow Estimation with Deep Networks](https://mlsanity.com/paper/1612.01925/) (2016)
- [DROID-SLAM: Deep Visual SLAM for Monocular, Stereo, and RGB-D Cameras](https://mlsanity.com/paper/2108.10869/) (2021)
- [LightGlue: Local Feature Matching at Light Speed](https://mlsanity.com/paper/2306.13643/) (2023)
- [TartanAir: A Dataset to Push the Limits of Visual SLAM](https://mlsanity.com/paper/2003.14338/) (2020)
- [Direct Sparse Odometry](https://mlsanity.com/paper/1607.02565/) (2016)
- [VINS-Mono: A Robust and Versatile Monocular Visual-Inertial State Estimator](https://mlsanity.com/paper/1708.03852/) (2017)
- [DISK: Learning local features with policy gradient](https://mlsanity.com/paper/2006.13566/) (2020)
- [HPatches: A benchmark and evaluation of handcrafted and learned local descriptors](https://mlsanity.com/paper/1704.05939/) (2017)
- [R2D2: Repeatable and Reliable Detector and Descriptor](https://mlsanity.com/paper/1906.06195/) (2019)
- [DeepVO: Towards End-to-End Visual Odometry with Deep Recurrent Convolutional Neural Networks](https://mlsanity.com/paper/1709.08429/) (2017)
- [BA-Net: Dense Bundle Adjustment Network](https://mlsanity.com/paper/1806.04807/) (2018)
- [SEA-RAFT: Simple, Efficient, Accurate RAFT for Optical Flow](https://mlsanity.com/paper/2405.14793/) (2024)
- [Key.Net: Keypoint Detection by Handcrafted and Learned CNN Filters](https://mlsanity.com/paper/1904.00889/) (2019)
- [XFeat: Accelerated Features for Lightweight Image Matching](https://mlsanity.com/paper/2404.19174/) (2024)
- [TartanVO: A Generalizable Learning-based VO](https://mlsanity.com/paper/2011.00359/) (2020)
- [ALIKE: Accurate and Lightweight Keypoint Detection and Descriptor Extraction](https://mlsanity.com/paper/2112.02906/) (2021)
- [Theseus: A Library for Differentiable Nonlinear Optimization](https://mlsanity.com/paper/2207.09442/) (2022)
- [PL-SLAM: a Stereo SLAM System through the Combination of Points and Line Segments](https://mlsanity.com/paper/1705.09479/) (2017)
- [Visual-Inertial Mapping with Non-Linear Factor Recovery](https://mlsanity.com/paper/1904.06504/) (2019)
- [PyPose: A Library for Robot Learning with Physics-based Optimization](https://mlsanity.com/paper/2209.15428/) (2022)
- [SFD2: Semantic-guided Feature Detection and Description](https://mlsanity.com/paper/2304.14845/) (2023)
- [AirSLAM: An Efficient and Illumination-Robust Point-Line Visual SLAM System](https://mlsanity.com/paper/2408.03520/) (2024)
- [An Evaluation of Feature Matchers for Fundamental Matrix Estimation](https://mlsanity.com/paper/1908.09474/) (2019)
- [AQUALOC: An Underwater Dataset for Visual-Inertial-Pressure Localization](https://mlsanity.com/paper/1910.14532/) (2019)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)

## Abstract (arXiv)

Sparse optical flow provides stable inter-frame correspondence, playing a key role in Visual Odometry (VO) and Visual-Inertial Odometry (VIO). Classical optimization-based methods, such as Lucas-Kanade (LK), perform well under small displacements but are sensitive to large motions and illumination changes. Modern regression-based learning methods, while more robust in complex scenes, are often computationally heavy and lack explicit geometric consistency, making them less suitable for efficient VO/VIO front-ends. To bridge this gap, we propose a hybrid neuro-symbolic framework that combines the strengths of both paradigms. Our method uses a Convolutional Neural Network (CNN) to extract robust feature representations, which is fed into a differentiable LK optimizer to estimate optical flow in an end-to-end trainable manner. Through implicit differentiation, gradients are propagated across the iterative solver, enabling joint optimization of feature extraction and flow estimation. The resulting system integrates seamlessly into existing VO/VIO pipelines and runs in real-time on embedded platforms. Experiments show that our method outperforms conventional optimization-based flow in challenging conditions such as dynamic lighting and low texture, while also achieving higher accuracy and lower latency than purely regression-based alternatives. When deployed in a VIO system, our method demonstrates significant performance improvement, achieving an average error reduction of 42% on challenging datasets while enhancing tracking stability. The code is publicly available.