# Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

arXiv: https://arxiv.org/abs/2007.11048 · HTML: https://mlsanity.com/paper/2007.11048/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2020.

## Built on by (most cited first, 10 of 10)

- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Neural parameter calibration for large-scale multi-agent models](https://mlsanity.com/paper/2209.13565/) (2022)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Learning stochastic dynamics and predicting emergent behavior using transformers](https://mlsanity.com/paper/2202.08708/) (2022)
- [Learning particle swarming models from data with Gaussian processes](https://mlsanity.com/paper/2106.02735/) (2021)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [Learning Nonlinear Couplings in Network of Agents from a Single Sample Trajectory](https://mlsanity.com/paper/2211.11069/) (2022)
- [Minimax Rates for Learning Pairwise Interactions in Attention-Style Models](https://mlsanity.com/paper/2510.11789/) (2025)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)

## Abstract (arXiv)

This paper concerns the parameter estimation problem for the quadratic potential energy in interacting particle systems from continuous-time and single-trajectory data. Even though such dynamical systems are high-dimensional, we show that the vanilla maximum likelihood estimator (without regularization) is able to estimate the interaction potential parameter with optimal rate of convergence simultaneously in mean-field limit and in long-time dynamics. This to some extend avoids the curse-of-dimensionality for estimating large dynamical systems under symmetry of the particle interaction.