# Learning Interaction Kernels from Collective Steady States

arXiv: https://arxiv.org/abs/2609.12004 · HTML: https://mlsanity.com/paper/2609.12004/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [One-shot learning for solution operators of partial differential equations](https://mlsanity.com/paper/2104.05512/) (2021)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems](https://mlsanity.com/paper/2010.03729/) (2020)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Data-driven Discovery of Emergent Behaviors in Collective Dynamics](https://mlsanity.com/paper/1912.11123/) (2019)
- [Learning Interaction Kernels for Agent Systems on Riemannian Manifolds](https://mlsanity.com/paper/2102.00327/) (2021)
- [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)
- [Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel](https://mlsanity.com/paper/2402.08412/) (2024)
- [Learning interacting particle systems from unlabeled data](https://mlsanity.com/paper/2604.02581/) (2026)
- [Learning Multi-type heterogeneous interacting particle systems](https://mlsanity.com/paper/2602.03954/) (2026)
- [Simultaneous inference of environmental and interaction forces in collective dynamics](https://mlsanity.com/paper/2608.25181/) (2026)

## Abstract (arXiv)

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.