In this paper, we focus on the data-driven discovery of a general second-order particle-based model that contains many state-of-the-art models for modeling the aggregation and collective behavior of interacting agents of similar size and body type. This model takes the form of a high-dimensional system of ordinary differential equations parameterized by two interaction kernels that appraise the alignment of positions and velocities. We propose a Gaussian Process-based approach to this problem, where the unknown model parameters are marginalized by using two independent Gaussian Process (GP) priors on latent interaction kernels constrained to dynamics and observational data. This results in a nonparametric model for interacting dynamical systems that accounts for uncertainty quantification. We also develop acceleration techniques to improve scalability. Moreover, we perform a theoretical analysis to interpret the methodology and investigate the conditions under which the kernels can be recovered. We demonstrate the effectiveness of the proposed approach on various prototype systems, including the selection of the order of the systems and the types of interactions. In particular, we present applications to modeling two real-world fish motion datasets that display flocking and milling patterns up to 248 dimensions. Despite the use of small data sets, the GP-based approach learns an effective representation of the nonlinear dynamics in these spaces and outperforms competitor methods.

翻译：本文聚焦于数据驱动发现一种通用的二阶基于粒子的模型，该模型包含许多用于建模相似尺寸和体型交互主体聚集与集体行为的最先进模型。该模型采用由两个相互作用核参数化的高维常微分方程组形式，这两个核用于评估位置和速度的对齐。我们提出了一种基于高斯过程的方法来解决该问题，其中通过使用两个独立的高斯过程先验对潜相互作用核进行边缘化，并约束于动力学和观测数据，从而得到用于交互动力系统的非参数模型，该模型考虑了不确定性量化。我们还开发了加速技术以提高可扩展性。此外，我们进行了理论分析以解释该方法，并研究了核可被恢复的条件。我们在各种原型系统上展示了所提出方法的有效性，包括系统阶次和交互类型的选择。特别地，我们将该方法应用于建模两个真实鱼类运动数据集，这些数据集展示了高达248维的集群和环游模式。尽管使用了小规模数据集，基于高斯过程的方法在这些空间中学习到了非线性动力学的有效表示，并且性能优于竞争方法。