# Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems

arXiv: https://arxiv.org/abs/2010.03729 · HTML: https://mlsanity.com/paper/2010.03729/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Oct 2020.

## Built on by (most cited first, 15 of 15)

- [Random Feature Models for Learning Interacting Dynamical Systems](https://mlsanity.com/paper/2212.05591/) (2022)
- [Learning Interaction Kernels for Agent Systems on Riemannian Manifolds](https://mlsanity.com/paper/2102.00327/) (2021)
- [Universal Approximation of Mean-Field Models via Transformers](https://mlsanity.com/paper/2410.16295/) (2024)
- [Learning Interaction Variables and Kernels from Observations of Agent-Based Systems](https://mlsanity.com/paper/2208.02758/) (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)
- [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)
- [Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel](https://mlsanity.com/paper/2402.08412/) (2024)
- [Machine Learning for Discovering Effective Interaction Kernels between Celestial Bodies from Ephemerides](https://mlsanity.com/paper/2108.11894/) (2021)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Model](https://mlsanity.com/paper/2505.07068/) (2025)
- [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)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (2026)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)

## Abstract (arXiv)

Modeling the complex interactions of systems of particles or agents is a fundamental scientific and mathematical problem that is studied in diverse fields, ranging from physics and biology, to economics and machine learning. In this work, we describe a very general second-order, heterogeneous, multivariable, interacting agent model, with an environment, that encompasses a wide variety of known systems. We describe an inference framework that uses nonparametric regression and approximation theory based techniques to efficiently derive estimators of the interaction kernels which drive these dynamical systems. We develop a complete learning theory which establishes strong consistency and optimal nonparametric min-max rates of convergence for the estimators, as well as provably accurate predicted trajectories. The estimators exploit the structure of the equations in order to overcome the curse of dimensionality and we describe a fundamental coercivity condition on the inverse problem which ensures that the kernels can be learned and relates to the minimal singular value of the learning matrix. The numerical algorithm presented to build the estimators is parallelizable, performs well on high-dimensional problems, and is demonstrated on complex dynamical systems.