# Learning interacting particle systems from unlabeled data

arXiv: https://arxiv.org/abs/2604.02581 · HTML: https://mlsanity.com/paper/2604.02581/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 44 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 15 of 15)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Proximal Optimal Transport Modeling of Population Dynamics](https://mlsanity.com/paper/2106.06345/) (2021)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Particle-based Energetic Variational Inference](https://mlsanity.com/paper/2004.06443/) (2020)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data](https://mlsanity.com/paper/2007.11048/) (2020)
- [Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation](https://mlsanity.com/paper/2106.13751/) (2021)
- [Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow](https://mlsanity.com/paper/2410.00229/) (2024)
- [Inverse Problems Over Probability Measure Space](https://mlsanity.com/paper/2504.18999/) (2025)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (2024)

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

- [Wasserstein Residuals: Learning Gradient Flows from Population Dynamics](https://mlsanity.com/paper/2607.04738/) (2026)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)

## Abstract (arXiv)

Learning the potentials of interacting particle systems is a fundamental task across various scientific disciplines. A major challenge is that unlabeled data collected at discrete time points lack trajectory information due to limitations in data collection methods or privacy constraints. We address this challenge by introducing a trajectory-free self-test loss function that leverages the weak-form stochastic evolution equation of the empirical distribution. The loss function is quadratic in potentials, supporting parametric and nonparametric regression algorithms for robust estimation that scale to large, high-dimensional systems with big data. Systematic numerical tests show that our method outperforms baseline methods that regress on trajectories recovered via label matching, tolerating large observation time steps. We establish the convergence of parametric estimators as the sample size increases, providing a theoretical foundation for the proposed approach.