# Flow Equivariant Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2507.14793 · HTML: https://mlsanity.com/paper/2507.14793/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 69 references that are in the map.
- First posted on arXiv 20 Jul 2025.

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

- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [4D Spatio-Temporal ConvNets: Minkowski Convolutional Neural Networks](https://mlsanity.com/paper/1904.08755/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Recurrent Vision Transformers for Object Detection with Event Cameras](https://mlsanity.com/paper/2212.05598/) (2022)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [MotionRNN: A Flexible Model for Video Prediction with Spacetime-Varying Motions](https://mlsanity.com/paper/2103.02243/) (2021)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Equi-Tuning: Group Equivariant Fine-Tuning of Pretrained Models](https://mlsanity.com/paper/2210.06475/) (2022)
- [Equivariant Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2408.14336/) (2024)
- [Traveling Waves Encode the Recent Past and Enhance Sequence Learning](https://mlsanity.com/paper/2309.08045/) (2023)
- [Rotation Equivariant Siamese Networks for Tracking](https://mlsanity.com/paper/2012.13078/) (2020)
- [Learning to Convolve: A Generalized Weight-Tying Approach](https://mlsanity.com/paper/1905.04663/) (2019)
- [A Spatiotemporal Perspective on Dynamical Computation in Neural Information Processing Systems](https://mlsanity.com/paper/2409.13669/) (2024)
- [Motion Equivariant Networks for Event Cameras with the Temporal Normalization Transform](https://mlsanity.com/paper/1902.06820/) (2019)
- [Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2311.10543/) (2023)
- [Recurrent Spatial Transformer Networks](https://mlsanity.com/paper/1509.05329/) (2015)
- [Covariant spatio-temporal receptive fields for spiking neural networks](https://mlsanity.com/paper/2405.00318/) (2024)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [A Prospective Study on Sequence-Driven Temporal Sampling and Ego-Motion Compensation for Action Recognition in the EPIC-Kitchens Dataset](https://mlsanity.com/paper/2008.11588/) (2020)
- [Observer Design for Nonlinear Systems with Equivariance](https://mlsanity.com/paper/2108.09387/) (2021)
- [Representing stimulus motion with waves in adaptive neural fields](https://mlsanity.com/paper/2312.06100/) (2023)
- [Task-Optimized Convolutional Recurrent Networks Align with Tactile Processing in the Rodent Brain](https://mlsanity.com/paper/2505.18361/) (2025)

## Built on by (most cited first, 1 of 1)

- [Flow Equivariant World Models: Memory for Partially Observed Dynamic Environments](https://mlsanity.com/paper/2601.01075/) (2026)

## Abstract (arXiv)

Data arrives at our senses as a continuous stream, smoothly transforming from one instant to the next. These smooth transformations can be viewed as continuous symmetries of the environment that we inhabit, defining equivalence relations between stimuli over time. In machine learning, neural network architectures that respect symmetries of their data are called equivariant and have provable benefits in terms of generalization ability and sample efficiency. To date, however, equivariance has been considered only for static transformations and feed-forward networks, limiting its applicability to sequence models, such as recurrent neural networks (RNNs), and corresponding time-parameterized sequence transformations. In this work, we extend equivariant network theory to this regime of 'flows' -- one-parameter Lie subgroups capturing natural transformations over time, such as visual motion. We begin by showing that standard RNNs are generally not flow equivariant: their hidden states fail to transform in a geometrically structured manner for moving stimuli. We then show how flow equivariance can be introduced, and demonstrate that these models significantly outperform their non-equivariant counterparts in terms of training speed, length generalization, and velocity generalization, on both next step prediction and sequence classification. We present this work as a first step towards building sequence models that respect the time-parameterized symmetries which govern the world around us.