# Identifiability of interaction kernels in mean-field equations of interacting particles

arXiv: https://arxiv.org/abs/2106.05565 · HTML: https://mlsanity.com/paper/2106.05565/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 40 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 7 of 7)

- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [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)
- [Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data](https://mlsanity.com/paper/2007.11048/) (2020)
- [On the coercivity condition in the learning of interacting particle systems](https://mlsanity.com/paper/2011.10480/) (2020)
- [Data-driven entropic spatially inhomogeneous evolutionary games](https://mlsanity.com/paper/2103.05429/) (2021)

## Built on by (most cited first, 12 of 12)

- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Data adaptive RKHS Tikhonov regularization for learning kernels in operators](https://mlsanity.com/paper/2203.03791/) (2022)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (2024)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures](https://mlsanity.com/paper/2311.00902/) (2023)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators](https://mlsanity.com/paper/2212.14163/) (2022)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [Unsupervised learning of observation functions in state-space models by nonparametric moment methods](https://mlsanity.com/paper/2207.05242/) (2022)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)
- [Fluctuations of Nonlinear Observables in Mean Field Neural Network Training](https://mlsanity.com/paper/2610.09768/) (2026)

## Abstract (arXiv)

This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive L2 spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each L2 space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted L2 space is preferable over the unweighted L2 space, as it yields more accurate regularized estimators.