# Soft Geometric Inductive Bias for Object Centric Dynamics

arXiv: https://arxiv.org/abs/2512.15493 · HTML: https://mlsanity.com/paper/2512.15493/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 35 references that are in the map.
- First posted on arXiv 17 Dec 2025.

## Builds on (most cited first, 30 of 33)

- [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)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Large Scale Interactive Motion Forecasting for Autonomous Driving : The Waymo Open Motion Dataset](https://mlsanity.com/paper/2104.10133/) (2021)
- [MONet: Unsupervised Scene Decomposition and Representation](https://mlsanity.com/paper/1901.11390/) (2019)
- [Multi-Object Representation Learning with Iterative Variational Inference](https://mlsanity.com/paper/1903.00450/) (2019)
- [Attend, Infer, Repeat: Fast Scene Understanding with Generative Models](https://mlsanity.com/paper/1603.08575/) (2016)
- [Contrastive Learning of Structured World Models](https://mlsanity.com/paper/1911.12247/) (2019)
- [SlotFormer: Unsupervised Visual Dynamics Simulation with Object-Centric Models](https://mlsanity.com/paper/2210.05861/) (2022)
- [Training Agents Inside of Scalable World Models](https://mlsanity.com/paper/2509.24527/) (2025)
- [Sequential Attend, Infer, Repeat: Generative Modelling of Moving Objects](https://mlsanity.com/paper/1806.01794/) (2018)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Deep Learning is Not So Mysterious or Different](https://mlsanity.com/paper/2503.02113/) (2025)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Physics-as-Inverse-Graphics: Unsupervised Physical Parameter Estimation from Video](https://mlsanity.com/paper/1905.11169/) (2019)
- [Kinetix: Investigating the Training of General Agents through Open-Ended Physics-Based Control Tasks](https://mlsanity.com/paper/2410.23208/) (2024)
- [Structured Object-Aware Physics Prediction for Video Modeling and Planning](https://mlsanity.com/paper/1910.02425/) (2019)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamics](https://mlsanity.com/paper/2501.18972/) (2025)
- [Regularizing Towards Soft Equivariance Under Mixed Symmetries](https://mlsanity.com/paper/2306.00356/) (2023)
- [Approximately Equivariant Graph Networks](https://mlsanity.com/paper/2308.10436/) (2023)
- [Improving Transformer World Models for Data-Efficient RL](https://mlsanity.com/paper/2502.01591/) (2025)
- [Graded Symmetry Groups: Plane and Simple](https://mlsanity.com/paper/2107.03771/) (2021)
- [Almost Equivariance via Lie Algebra Convolutions](https://mlsanity.com/paper/2310.13164/) (2023)
- [Platonic Transformers: A Solid Choice For Equivariance](https://mlsanity.com/paper/2510.03511/) (2025)

## Abstract (arXiv)

Physical systems are naturally parameterized in terms of geometric entities and their transformations. While exact equivariance can be a powerful inductive bias, real-world applications frequently exhibit approximate or broken symmetries in practice. We introduce object-centric world models that provide a soft geometric inductive bias without enforcing exact symmetries. By embedding object states as Clifford multivectors, our models encourage physically meaningful transformations while retaining the expressivity needed to handle asymmetric dynamics. We evaluate our approach on 2D rigid-body dynamics, 3D charged-particle systems, and real-world driving trajectories. Compared to both unstructured baselines and strictly equivariant models, our soft Clifford transformer achieves better long-horizon fidelity, particularly in regimes with broken symmetries. These results suggest that geometric algebra offers an effective middle ground, delivering sample-efficient dynamics without the inflexibility of hard mathematical constraints.