# Learning particle swarming models from data with Gaussian processes

arXiv: https://arxiv.org/abs/2106.02735 · HTML: https://mlsanity.com/paper/2106.02735/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 69 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 26 of 26)

- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [A general framework for Vecchia approximations of Gaussian processes](https://mlsanity.com/paper/1708.06302/) (2017)
- [Local Gaussian process approximation for large computer experiments](https://mlsanity.com/paper/1303.0383/) (2013)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [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 Stochastic Differential Equations With Gaussian Processes Without Gradient Matching](https://mlsanity.com/paper/1807.05748/) (2018)
- [Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes](https://mlsanity.com/paper/2009.07444/) (2020)
- [Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems](https://mlsanity.com/paper/2010.03729/) (2020)
- [Gaussian Process Assisted Active Learning of Physical Laws](https://mlsanity.com/paper/1910.03120/) (2019)
- [Data-driven Discovery of Emergent Behaviors in Collective Dynamics](https://mlsanity.com/paper/1912.11123/) (2019)
- [Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data](https://mlsanity.com/paper/2007.11048/) (2020)
- [Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice](https://mlsanity.com/paper/1402.4281/) (2014)
- [Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations](https://mlsanity.com/paper/1810.11596/) (2018)

## Built on by (most cited first, 3 of 3)

- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications](https://mlsanity.com/paper/2511.02053/) (2025)

## Abstract (arXiv)

Interacting particle or agent systems that display a rich variety of swarming behaviours are ubiquitous in science and engineering. A fundamental and challenging goal is to understand the link between individual interaction rules and swarming. In this paper, we study the data-driven discovery of a second-order particle swarming model that describes the evolution of N particles in ℝd under radial interactions. We propose a learning approach that models the latent radial interaction function as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of the interaction function with pointwise uncertainty quantification, and the other one is the inference of unknown scalar parameters in the non-collective friction forces of the system. We formulate the learning problem as a statistical inverse problem and provide a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. Given data collected from M i.i.d trajectories with independent Gaussian observational noise, we provide a finite-sample analysis, showing that our posterior mean estimator converges in a Reproducing kernel Hilbert space norm, at an optimal rate in M equal to the one in the classical 1-dimensional Kernel Ridge regression. As a byproduct, we show we can obtain a parametric learning rate in M for the posterior marginal variance using L∞ norm, and the rate could also involve N and L (the number of observation time instances for each trajectory), depending on the condition number of the inverse problem. Numerical results on systems that exhibit different swarming behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.