用几何结构提升物理系统建模的泛化能力
Geometric Principles for Machine Learning of Dynamical Systems
- 基于对称正定流形构建几何感知的机器学习框架
- 在低维线性系统上实现优于传统模型的结构泛化性能
- 适合研究物理信息机器学习与几何深度学习的学者
动力系统的形式化描述深深植根于由非欧几何定义的拓扑空间。本文提出利用结构丰富的几何空间进行机器学习，以实现从数据中建模物理系统时的结构泛化，而非将物理先验嵌入无模型架构中。我们将模型泛化视为对称性、不变性和唯一性的函数，定义为状态空间动力学到参数空间的拓扑映射。通过线性时不变动力系统的机器学习实例说明这一观点，其动力学位于对称正定流形上。
原文摘要 · Abstract (English)
Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural generalization when modeling physical systems from data, in contrast to embedding physics bias within model-free architectures. We consider model generalization to be a function of symmetry, invariance and uniqueness, defined as a topological mapping from state space dynamics to the parameter space. We illustrate this view through the machine learning of linear time-invariant dynamical systems, whose dynamics reside on the symmetric positive definite manifold.
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