Learning particle swarming models from data with Gaussian processes

2021/06/04 by Jinchao Feng, Feng, Jinchao, Kulick, Charles +4 · 3 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Protein Structure and Dynamics #Statistical Mechanics and Entropy #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2106.02735

openalex publication_date 2021/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ℝ^d 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.

Citations

• On the limited memory BFGS method for large scale optimization
• Heterophilious Dynamics Enhances Consensus
• Frequentist coverage and sup-norm convergence rate in Gaussian process regression
• Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences
• Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories
• Solving and Learning Nonlinear PDEs with Gaussian Processes
• On the Estimation of Derivatives Using Plug-in Kernel Ridge Regression Estimators

Cited by

• A kernel method for the learning of Wasserstein geometric flows
• Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications
• Automatic reproducing kernel and regularization for learning convolution kernels

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