从嵌入到控制：随机多目标系统的表示方法
From Embedding to Control: Representations for Stochastic Multi-Object Systems
AI总结：
针对随机非线性多目标系统的建模与控制难题，提出基于希尔伯特空间嵌入的图可控嵌入框架GCE，结合平均场近似与图神经网络实现高效建模与线性控制，在多领域实验中性能优于现有嵌入方法。
摘要：
本文研究如何在具有多个相互作用对象的随机非线性动力学中实现精确建模与有效控制。然而，非均匀相互作用和随机拓扑结构使该任务颇具挑战性。针对这些挑战，我们提出了\textit{Graph Controllable Embeddings}（图可控嵌入，简称GCE）——一种用于学习随机多目标动力学以实现线性控制的通用框架。具体而言，GCE基于希尔伯特空间嵌入构建，可将受控随机动力学的概率分布直接嵌入到再生核希尔伯特空间（RKHS）中，从而在保留非线性表达能力的同时，在其RKHS内实现线性运算。我们为GCE的存在性、收敛性和适用性提供了理论保证。值得注意的是，该框架采用平均场近似技术，可高效捕获对象间依赖关系，并实现可证明的低样本复杂度。通过整合图神经网络，我们构建了依赖于数据的核特征，这些特征能够适应动态交互模式，且仅需有限训练样本即可泛化至未见过的拓扑结构。GCE可无缝扩展至不同规模和拓扑的多目标系统。借助希尔伯特空间的线性特性，GCE还支持用于合成最优序列的简单且有效的控制算法。在物理系统、机器人技术和电网领域的实验验证了GCE的有效性，并表明在分布内测试和少样本测试中，GCE的性能均始终优于多种主流嵌入方法。
英文摘要：
This paper studies how to achieve accurate modeling and effective control in stochastic nonlinear dynamics with multiple interacting objects. However, non-uniform interactions and random topologies make this task challenging. We address these challenges by proposing \textit{Graph Controllable Embeddings} (GCE), a general framework to learn stochastic multi-object dynamics for linear control. Specifically, GCE is built on Hilbert space embeddings, allowing direct embedding of probability distributions of controlled stochastic dynamics into a reproducing kernel Hilbert space (RKHS), which enables linear operations in its RKHS while retaining nonlinear expressiveness. We provide theoretical guarantees on the existence, convergence, and applicability of GCE. Notably, a mean field approximation technique is adopted to efficiently capture inter-object dependencies and achieve provably low sample complexity. By integrating graph neural networks, we construct data-dependent kernel features that are capable of adapting to dynamic interaction patterns and generalizing to even unseen topologies with only limited training instances. GCE scales seamlessly to multi-object systems of varying sizes and topologies. Leveraging the linearity of Hilbert spaces, GCE also supports simple yet effective control algorithms for synthesizing optimal sequences. Experiments on physical systems, robotics, and power grids validate GCE and demonstrate consistent performance improvement over various competitive embedding methods in both in-distribution and few-shot tests
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
