This paper studies the problem of modeling interacting dynamical systems,
which is critical for understanding physical dynamics and biological processes.
Recent research predominantly uses geometric graphs to represent these
interactions, which are then captured by powerful graph neural networks (GNNs).
However, predicting interacting dynamics in challenging scenarios such as
out-of-distribution shift and complicated underlying rules remains unsolved. In
this paper, we propose a new approach named Graph ODE with factorized
prototypes (GOAT) to address the problem. The core of GOAT is to incorporate
factorized prototypes from contextual knowledge into a continuous graph ODE
framework. Specifically, GOAT employs representation disentanglement and system
parameters to extract both object-level and system-level contexts from
historical trajectories, which allows us to explicitly model their independent
influence and thus enhances the generalization capability under system changes.
Then, we integrate these disentangled latent representations into a graph ODE
model, which determines a combination of various interacting prototypes for
enhanced model expressivity. The entire model is optimized using an end-to-end
variational inference framework to maximize the likelihood. Extensive
experiments in both in-distribution and out-of-distribution settings validate
the superiority of GOAT.

通过引入因子化的原型，使用图形 ODE 模型和变分推论框架，提出了一种名为 GOAT 的新方法，用于建模相互作用的动态系统，以增强模型的推广能力。

图 ODE 与因式原型的建模复杂交互动力学

Graph ODE with Factorized Prototypes for Modeling Complicated  Interacting Dynamics

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.

本文通过数据驱动的方法，提出了一种基于粒子的模型，使用基于高斯过程的方法来对相互作用的动力学系统进行建模，并利用小型数据集证明了该方法在描述非线性动力学系统方面的有效性。

基于高斯过程和低维交互结构的二阶粒子动力学数据驱动模型选择

Data-Driven Model Selections of Second-Order Particle Dynamics via  Integrating Gaussian Processes with Low-Dimensional Interacting Structures

The study of interacting dynamical systems continues to attract research
interest in various fields of science and engineering. In a collection of
interacting particles, the interaction network contains information about how
various components interact with one another. Inferring the information about
the interaction network from the dynamics of agents is a problem of
long-standing interest. In this work, we employ a self-supervised neural
network model to achieve two outcomes: to recover the interaction network and
to predict the dynamics of individual agents. Both these information are
inferred solely from the observed trajectory data. This work presents an
application of the Neural Relational Inference model to two dynamical systems:
coupled particles mediated by Hooke's law interaction and coupled phase
(Kuramoto) oscillators.

利用自监督神经网络模型从观察到的轨迹数据中恢复相互作用网络并预测个体动力学的研究，应用于耦合粒子和耦合振荡器（Kuramoto）的两个动力系统。