# PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling

arXiv: https://arxiv.org/abs/2510.04233 · HTML: https://mlsanity.com/paper/2510.04233/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 63 references that are in the map.
- First posted on arXiv 5 Oct 2025.

## Builds on (most cited first, 30 of 32)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (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)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Learning Gradient Fields for Molecular Conformation Generation](https://mlsanity.com/paper/2105.03902/) (2021)
- [A Unified View on Graph Neural Networks as Graph Signal Denoising](https://mlsanity.com/paper/2010.01777/) (2020)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Flexible Neural Representation for Physics Prediction](https://mlsanity.com/paper/1806.08047/) (2018)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?](https://mlsanity.com/paper/2410.11443/) (2024)
- [Geometric Trajectory Diffusion Models](https://mlsanity.com/paper/2410.13027/) (2024)
- [Force-Free Molecular Dynamics Through Autoregressive Equivariant Networks](https://mlsanity.com/paper/2503.23794/) (2025)
- [Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling](https://mlsanity.com/paper/2410.06366/) (2024)
- [Spatio-Temporal Fluid Dynamics Modeling via Physical-Awareness and Parameter Diffusion Guidance](https://mlsanity.com/paper/2403.13850/) (2024)

## Built on by (most cited first, 2 of 2)

- [Generative Modeling in Protein Design: Neural Representations, Conditional Generation, and Evaluation Standards](https://mlsanity.com/paper/2603.26378/) (2026)
- [ExPhy: A Benchmark for Explicit Physical Property Learning in Multi-Object Trajectory Forecasting](https://mlsanity.com/paper/2608.20009/) (2026)

## Abstract (arXiv)

Modeling 3D dynamics is a fundamental problem in multi-body systems across scientific and engineering domains and has important practical implications in object trajectory prediction and simulation. While recent GNN-based approaches have achieved strong performance by enforcing geometric symmetries, encoding high-order features or incorporating neural-ODE mechanics, they typically depend on explicitly observed structures and inherently fail to capture the unobserved interactions that are crucial to complex physical behaviors and dynamics mechanism. In this paper, we propose PAINET, a principled SE(3)-equivariant transformer for learning all-pair interactions in multi-body systems. The model comprises: (1) a novel physics-inspired attention network derived from the minimization trajectory of an energy function, and (2) a parallel decoder that preserves equivariance while enabling efficient inference. Empirical results on diverse real-world benchmarks, including human motion capture, molecular dynamics, and large-scale protein simulations, show that PAINET consistently outperforms recently proposed models, yielding 4.7% to 41.5% error reductions in 3D dynamics prediction with comparable computation costs in terms of time and memory. Our codes, baseline models and datasets are available at https://github.com/Icarus1411/PAINET.