# Learning Permutation-invariant Macroscopic Dynamics

arXiv: https://arxiv.org/abs/2605.30812 · HTML: https://mlsanity.com/paper/2605.30812/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 58 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 30 of 36)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Attention-based Deep Multiple Instance Learning](https://mlsanity.com/paper/1802.04712/) (2018)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [AtlasNet: A Papier-Mach'e Approach to Learning 3D Surface Generation](https://mlsanity.com/paper/1802.05384/) (2018)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [DN-DETR: Accelerate DETR Training by Introducing Query DeNoising](https://mlsanity.com/paper/2203.01305/) (2022)
- [PU-Net: Point Cloud Upsampling Network](https://mlsanity.com/paper/1801.06761/) (2018)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [3D Point Capsule Networks](https://mlsanity.com/paper/1812.10775/) (2018)
- [Deep Set Prediction Networks](https://mlsanity.com/paper/1906.06565/) (2019)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Understanding Self-Predictive Learning for Reinforcement Learning](https://mlsanity.com/paper/2212.03319/) (2022)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations](https://mlsanity.com/paper/2107.04556/) (2021)
- [Which priors matter? Benchmarking models for learning latent dynamics](https://mlsanity.com/paper/2111.05458/) (2021)
- [Rep the Set: Neural Networks for Learning Set Representations](https://mlsanity.com/paper/1904.01962/) (2019)
- [Constructing Custom Thermodynamics Using Deep Learning](https://mlsanity.com/paper/2308.04119/) (2023)
- [Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems](https://mlsanity.com/paper/2301.10391/) (2023)
- [Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations](https://mlsanity.com/paper/2306.01174/) (2023)
- [Progressive Seed Generation Auto-encoder for Unsupervised Point Cloud Learning](https://mlsanity.com/paper/2112.05213/) (2021)
- [Learn to Predict Sets Using Feed-Forward Neural Networks](https://mlsanity.com/paper/2001.11845/) (2020)
- [Simplifying Latent Dynamics with Softly State-Invariant World Models](https://mlsanity.com/paper/2401.17835/) (2024)

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

- [Learning Macroscopic Dynamics without Reconstructing Microscopic States](https://mlsanity.com/paper/2609.37392/) (2026)

## Abstract (arXiv)

Accurately modeling the macroscopic dynamics of high-dimensional microscopic systems is of broad interest across the sciences. Many data-driven approaches learn a low-dimensional latent state through an autoencoder trained for pointwise input reconstruction. These methods typically assume a fixed ordering of microscopic degrees of freedom in the input. However, in many settings, such as particle systems, the microscopic state is inherently unordered. This motivates an autoencoder framework that learns permutation-invariant latent representations. To this end, we adopt a permutation-invariant encoder and design the decoder to reconstruct the mass distribution centered at the observed points rather than per-sample reconstruction. We then jointly learn the macroscopic dynamics of the observables together with the latent states. We demonstrate the effectiveness and robustness of the proposed method across a range of microscopic settings, including learning the energy dynamics in interacting particle systems, predicting mixing dynamics in Lennard-Jones fluids, and modeling the stretching dynamics from video data of polymers moving in an elongational force field.