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.

翻译：我们提出一个框架，用于从多轨迹数据中联合推断异构交互粒子系统的网络拓扑、多类型交互核与潜在类型分配。该学习任务是一个具有挑战性的非凸混合整数优化问题，我们通过一种新颖的三阶段方法加以解决。首先，我们利用智能体交互间的共享结构，通过矩阵感知恢复系统参数的低秩嵌入。其次，我们在学习到的嵌入空间内通过聚类识别离散的交互类型。最后，我们通过矩阵分解和后处理细化来恢复网络权重矩阵与核系数。我们在受限等距性质（RIP）假设下提供了带有估计误差界的理论保证，并基于聚类可分性建立了精确恢复交互类型的条件。在合成数据集（包括异构捕食者-猎物系统）上的数值实验表明，我们的方法能够准确重建底层动力学，并对噪声具有鲁棒性。