# Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2102.00327 · HTML: https://mlsanity.com/paper/2102.00327/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 52 references that are in the map.
- First posted on arXiv 30 Jan 2021.

## Builds on (most cited first, 14 of 14)

- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [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)
- [Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems](https://mlsanity.com/paper/1403.4290/) (2014)
- [Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes](https://mlsanity.com/paper/2009.07444/) (2020)
- [Discovery of Dynamics Using Linear Multistep Methods](https://mlsanity.com/paper/1912.12728/) (2019)
- [Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems](https://mlsanity.com/paper/2010.03729/) (2020)
- [Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets](https://mlsanity.com/paper/2010.14324/) (2020)
- [Data-driven Discovery of Emergent Behaviors in Collective Dynamics](https://mlsanity.com/paper/1912.11123/) (2019)
- [Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks](https://mlsanity.com/paper/2011.01797/) (2020)
- [Exploring the coevolution of predator and prey morphology and behavior](https://mlsanity.com/paper/1602.08802/) (2016)
- [DOP: Deep Optimistic Planning with Approximate Value Function Evaluation](https://mlsanity.com/paper/1803.08501/) (2018)
- [Multi-Agent Systems based on Contextual Defeasible Logic considering Focus](https://mlsanity.com/paper/2010.00168/) (2020)

## Built on by (most cited first, 7 of 7)

- [An Empirical Study of Agent Developer Practices in AI Agent Frameworks](https://mlsanity.com/paper/2512.01939/) (2025)
- [Learning Interaction Variables and Kernels from Observations of Agent-Based Systems](https://mlsanity.com/paper/2208.02758/) (2022)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Machine Learning for Discovering Effective Interaction Kernels between Celestial Bodies from Ephemerides](https://mlsanity.com/paper/2108.11894/) (2021)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [GRAPHYP: A Scientific Knowledge Graph with Manifold Subnetworks of Communities. Detection of Scholarly Disputes in Adversarial Information Routes](https://mlsanity.com/paper/2205.01331/) (2022)
- [Learning Interaction Kernels from Collective Steady States](https://mlsanity.com/paper/2609.12004/) (2026)

## Abstract (arXiv)

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the 2-dimensional sphere and the Poincar'e disk model of hyperbolic space.