# EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoning

arXiv: https://arxiv.org/abs/2303.10876 · HTML: https://mlsanity.com/paper/2303.10876/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 79 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Builds on (most cited first, 30 of 59)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Social GAN: Socially Acceptable Trajectories with Generative Adversarial Networks](https://mlsanity.com/paper/1803.10892/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [VectorNet: Encoding HD Maps and Agent Dynamics from Vectorized Representation](https://mlsanity.com/paper/2005.04259/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [MultiPath: Multiple Probabilistic Anchor Trajectory Hypotheses for Behavior Prediction](https://mlsanity.com/paper/1910.05449/) (2019)
- [Learning Lane Graph Representations for Motion Forecasting](https://mlsanity.com/paper/2007.13732/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [On human motion prediction using recurrent neural networks](https://mlsanity.com/paper/1705.02445/) (2017)
- [AgentFormer: Agent-Aware Transformers for Socio-Temporal Multi-Agent Forecasting](https://mlsanity.com/paper/2103.14023/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [DESIRE: Distant Future Prediction in Dynamic Scenes with Interacting Agents](https://mlsanity.com/paper/1704.04394/) (2017)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Learning Trajectory Dependencies for Human Motion Prediction](https://mlsanity.com/paper/1908.05436/) (2019)
- [It Is Not the Journey but the Destination: Endpoint Conditioned Trajectory Prediction](https://mlsanity.com/paper/2004.02025/) (2020)
- [Structural-RNN: Deep Learning on Spatio-Temporal Graphs](https://mlsanity.com/paper/1511.05298/) (2015)
- [Actional-Structural Graph Convolutional Networks for Skeleton-based Action Recognition](https://mlsanity.com/paper/1904.12659/) (2019)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Social Attention: Modeling Attention in Human Crowds](https://mlsanity.com/paper/1710.04689/) (2017)
- [Stochastic Trajectory Prediction via Motion Indeterminacy Diffusion](https://mlsanity.com/paper/2203.13777/) (2022)
- [Transformer Networks for Trajectory Forecasting](https://mlsanity.com/paper/2003.08111/) (2020)
- [History Repeats Itself: Human Motion Prediction via Motion Attention](https://mlsanity.com/paper/2007.11755/) (2020)
- [IntentNet: Learning to Predict Intention from Raw Sensor Data](https://mlsanity.com/paper/2101.07907/) (2021)
- [Dynamic Multiscale Graph Neural Networks for 3D Skeleton-Based Human Motion Prediction](https://mlsanity.com/paper/2003.08802/) (2020)
- [Distilling Knowledge from Graph Convolutional Networks](https://mlsanity.com/paper/2003.10477/) (2020)

## Built on by (most cited first, 30 of 97)

- [SingularTrajectory: Universal Trajectory Predictor Using Diffusion Model](https://mlsanity.com/paper/2403.18452/) (2024)
- [Social-Transmotion: Promptable Human Trajectory Prediction](https://mlsanity.com/paper/2312.16168/) (2023)
- [Can Language Beat Numerical Regression? Language-Based Multimodal Trajectory Prediction](https://mlsanity.com/paper/2403.18447/) (2024)
- [MoFlow: One-Step Flow Matching for Human Trajectory Forecasting via Implicit Maximum Likelihood Estimation based Distillation](https://mlsanity.com/paper/2503.09950/) (2025)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [SocialCircle: Learning the Angle-based Social Interaction Representation for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2310.05370/) (2023)
- [Skeleton-in-Context: Unified Skeleton Sequence Modeling with In-Context Learning](https://mlsanity.com/paper/2312.03703/) (2023)
- [Clifford Group Equivariant Simplicial Message Passing Networks](https://mlsanity.com/paper/2402.10011/) (2024)
- [GuideFlow: Constraint-Guided Flow Matching for Planning in End-to-End Autonomous Driving](https://mlsanity.com/paper/2511.18729/) (2025)
- [EquiDiff: A Conditional Equivariant Diffusion Model For Trajectory Prediction](https://mlsanity.com/paper/2308.06564/) (2023)
- [MART: MultiscAle Relational Transformer Networks for Multi-agent Trajectory Prediction](https://mlsanity.com/paper/2407.21635/) (2024)
- [Multi-Transmotion: Pre-trained Model for Human Motion Prediction](https://mlsanity.com/paper/2411.02673/) (2024)
- [Auxiliary Tasks Benefit 3D Skeleton-based Human Motion Prediction](https://mlsanity.com/paper/2308.08942/) (2023)
- [T4P: Test-Time Training of Trajectory Prediction via Masked Autoencoder and Actor-specific Token Memory](https://mlsanity.com/paper/2403.10052/) (2024)
- [Language-Driven Interactive Traffic Trajectory Generation](https://mlsanity.com/paper/2405.15388/) (2024)
- [Sports-Traj: A Unified Trajectory Generation Model for Multi-Agent Movement in Sports](https://mlsanity.com/paper/2405.17680/) (2024)
- [LaneCPP: Continuous 3D Lane Detection using Physical Priors](https://mlsanity.com/paper/2406.08381/) (2024)
- [Joint-Relation Transformer for Multi-Person Motion Prediction](https://mlsanity.com/paper/2308.04808/) (2023)
- [Geometric Trajectory Diffusion Models](https://mlsanity.com/paper/2410.13027/) (2024)
- [Unified Uncertainty-Aware Diffusion for Multi-Agent Trajectory Modeling](https://mlsanity.com/paper/2503.18589/) (2025)
- [Another Vertical View: A Hierarchical Network for Heterogeneous Trajectory Prediction via Spectrums](https://mlsanity.com/paper/2304.05106/) (2023)
- [Trajectory Prediction for Autonomous Driving: Progress, Limitations, and Future Directions](https://mlsanity.com/paper/2503.03262/) (2025)
- [Unified Spatial-Temporal Edge-Enhanced Graph Networks for Pedestrian Trajectory Prediction](https://mlsanity.com/paper/2502.02504/) (2025)
- [Intention-aware Denoising Diffusion Model for Trajectory Prediction](https://mlsanity.com/paper/2403.09190/) (2024)
- [TranSPORTmer: A Holistic Approach to Trajectory Understanding in Multi-Agent Sports](https://mlsanity.com/paper/2410.17785/) (2024)
- [EgoCast: Forecasting Egocentric Human Pose in the Wild](https://mlsanity.com/paper/2412.02903/) (2024)
- [Physical Plausibility-aware Trajectory Prediction via Locomotion Embodiment](https://mlsanity.com/paper/2503.17267/) (2025)
- [Multi-modal Knowledge Distillation-based Human Trajectory Forecasting](https://mlsanity.com/paper/2503.22201/) (2025)
- [BP-SGCN: Behavioral Pseudo-Label Informed Sparse Graph Convolution Network for Pedestrian and Heterogeneous Trajectory Prediction](https://mlsanity.com/paper/2502.14676/) (2025)
- [Pedestrian Trajectory Prediction with Missing Data: Datasets, Imputation, and Benchmarking](https://mlsanity.com/paper/2411.00174/) (2024)

## Abstract (arXiv)

Learning to predict agent motions with relationship reasoning is important for many applications. In motion prediction tasks, maintaining motion equivariance under Euclidean geometric transformations and invariance of agent interaction is a critical and fundamental principle. However, such equivariance and invariance properties are overlooked by most existing methods. To fill this gap, we propose EqMotion, an efficient equivariant motion prediction model with invariant interaction reasoning. To achieve motion equivariance, we propose an equivariant geometric feature learning module to learn a Euclidean transformable feature through dedicated designs of equivariant operations. To reason agent's interactions, we propose an invariant interaction reasoning module to achieve a more stable interaction modeling. To further promote more comprehensive motion features, we propose an invariant pattern feature learning module to learn an invariant pattern feature, which cooperates with the equivariant geometric feature to enhance network expressiveness. We conduct experiments for the proposed model on four distinct scenarios: particle dynamics, molecule dynamics, human skeleton motion prediction and pedestrian trajectory prediction. Experimental results show that our method is not only generally applicable, but also achieves state-of-the-art prediction performances on all the four tasks, improving by 24.0/30.1/8.6/9.2%. Code is available at https://github.com/MediaBrain-SJTU/EqMotion.