# Permutation-equivariant neural networks applied to dynamics prediction

arXiv: https://arxiv.org/abs/1612.04530 · HTML: https://mlsanity.com/paper/1612.04530/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Dec 2016.

## Built on by (most cited first, 14 of 14)

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- [Why self-attention is Natural for Sequence-to-Sequence Problems? A Perspective from Symmetries](https://mlsanity.com/paper/2210.06741/) (2022)
- [Equivariant Deep Dynamical Model for Motion Prediction](https://mlsanity.com/paper/2111.01892/) (2021)
- [Circular-Symmetric Correlation Layer based on FFT](https://mlsanity.com/paper/2107.12480/) (2021)
- [REViT: Roto-reflection Equivariant Convolutional Vision Transformer](https://mlsanity.com/paper/2606.25318/) (2026)
- [Battery GraphNets : Relational Learning for Lithium-ion Batteries(LiBs) Life Estimation](https://mlsanity.com/paper/2408.07624/) (2024)
- [A Diagrammatic Approach to Improve Computational Efficiency in Group Equivariant Neural Networks](https://mlsanity.com/paper/2412.10837/) (2024)
- [Centralized Permutation Equivariant Policy for Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2508.11706/) (2025)

## Abstract (arXiv)

The introduction of convolutional layers greatly advanced the performance of neural networks on image tasks due to innately capturing a way of encoding and learning translation-invariant operations, matching one of the underlying symmetries of the image domain. In comparison, there are a number of problems in which there are a number of different inputs which are all 'of the same type' --- multiple particles, multiple agents, multiple stock prices, etc. The corresponding symmetry to this is permutation symmetry, in that the algorithm should not depend on the specific ordering of the input data. We discuss a permutation-invariant neural network layer in analogy to convolutional layers, and show the ability of this architecture to learn to predict the motion of a variable number of interacting hard discs in 2D. In the same way that convolutional layers can generalize to different image sizes, the permutation layer we describe generalizes to different numbers of objects.