# Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles

arXiv: https://arxiv.org/abs/2010.15694 · HTML: https://mlsanity.com/paper/2010.15694/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2020.

## Built on by (most cited first, 21 of 21)

- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [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)
- [Bayesian Nonlocal Operator Regression (BNOR): A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification](https://mlsanity.com/paper/2211.01330/) (2022)
- [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)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators](https://mlsanity.com/paper/2212.14163/) (2022)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Small noise analysis for Tikhonov and RKHS regularizations](https://mlsanity.com/paper/2305.11055/) (2023)
- [Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications](https://mlsanity.com/paper/2511.02053/) (2025)
- [Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems](https://mlsanity.com/paper/2603.15927/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)

## Abstract (arXiv)

We introduce a nonparametric algorithm to learn interaction kernels of mean-field equations for 1st-order systems of interacting particles. The data consist of discrete space-time observations of the solution. By least squares with regularization, the algorithm learns the kernel on data-adaptive hypothesis spaces efficiently. A key ingredient is a probabilistic error functional derived from the likelihood of the mean-field equation's diffusion process. The estimator converges, in a reproducing kernel Hilbert space and an L2 space under an identifiability condition, at a rate optimal in the sense that it equals the numerical integrator's order. We demonstrate our algorithm on three typical examples: the opinion dynamics with a piecewise linear kernel, the granular media model with a quadratic kernel, and the aggregation-diffusion with a repulsive-attractive kernel.