Learning Multi-type heterogeneous interacting particle systems

TL;DR

This work tackles the challenging problem of jointly identifying network topology, multi-type interaction kernels, and latent agent types in heterogeneous interacting particle systems from multiple trajectories. It introduces a novel three-stage framework that first recovers a low-rank kernel-graph embedding via matrix sensing, then clusters to infer interaction types, and finally factorizes to recover the network weights and kernel coefficients with an ALS-based refinement. Theoretical guarantees are provided under a Restricted Isometry Property and cluster-separability conditions, ensuring accurate recovery of the type assignments and a robust estimation of the dynamics. Numerical experiments on synthetic data, including heterogeneous predator-prey systems, demonstrate accurate reconstruction of the underlying dynamics and robustness to noise, highlighting the framework’s scalability and practical relevance for learning structured interactions in multi-agent systems.

Abstract

We propose a framework for the joint inference of network topology, multi-type interaction kernels, and latent type assignments in heterogeneous interacting particle systems from multi-trajectory data. This learning task is a challenging non-convex mixed-integer optimization problem, which we address through a novel three-stage approach. First, we leverage shared structure across agent interactions to recover a low-rank embedding of the system parameters via matrix sensing. Second, we identify discrete interaction types by clustering within the learned embedding. Third, we recover the network weight matrix and kernel coefficients through matrix factorization and a post-processing refinement. We provide theoretical guarantees with estimation error bounds under a Restricted Isometry Property (RIP) assumption and establish conditions for the exact recovery of interaction types based on cluster separability. Numerical experiments on synthetic datasets, including heterogeneous predator-prey systems, demonstrate that our method yields an accurate reconstruction of the underlying dynamics and is robust to noise.

Figures (8)

• Figure 1: From the type matrix $\boldsymbol{\kappa}$ to assignment matrices $\boldsymbol{\mathcal{K}}_1$ and $\boldsymbol{\mathcal{K}}_2$. The diagonal element of $\boldsymbol{\kappa}$ is not important since self-interactions are not allowed.
• Figure 2: Examples of row vectors from the matrix $\widehat{\mathbf{Z}}$ estimated from noisy data over 2D hypothesis space ($K=2$), with $Q=3$ types of nonzero rows. Type 0 corresponds to a $0$ entry on the graph. Left: All types are clearly distinguishable due to sufficiently large angle $\theta_{min}$ and row norm $z_{min}$. Middle: Nonzero rows of different types become indistinguishable due to $\theta_{\min}$ being small. Right: Zero and nonzero rows become indistinguishable due to $z_{\min}$ being small.
• Figure 3: Typical trajectory of a predator-prey system. Top: Observed trajectory data of 6 predators and 4 prey, with roles of individual particles unknown. Bottom: Ground-truth classification of predators and prey.
• Figure 4: Graph, kernel, and their estimations in the predator-prey system: predator-prey ($\phi_1$), prey-prey ($\phi_2$), prey-predator ($\phi_3$), and predator-predator ($\phi_4$).
• Figure 5: Convergence of estimation errors with sample size $M$. Left: dense graph. Right: sparse graph. Errors in $\mathbf{Z}$ and trajectory prediction decrease once $M > 32$, with the $\mathbf{Z}$ error decaying at the rate $M^{-1/2}$. Successful recovery of the type matrix $\kappa$ occurs when the $\mathbf{Z}$ error falls below the separation thresholds (around $M=300$ and $M = 54$), after which kernel errors also begin to decay. Sparse graphs require fewer samples for recovery than dense graphs.

Theorems & Definitions (27)

• Example 1.1: Heterogeneous interaction kernel systems
• Remark 1.2: Complete networks
• Remark 1.3: Kernel heterogeneity
• Example 2.1
• Example 2.2
• Remark 2.3: Complete network
• Remark 3.1: Transformation invariance
• Remark 3.2: Cluster ordering
• Remark 3.3: Active and inactive kernels
• Remark 3.4