发表机构
University of Cambridge; University of Bath（剑桥大学; 巴斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对从轨迹数据辨识分段光滑动力系统的问题，提出模块化框架，结合切换超平面估计与几何约束神经网络，在基准问题上验证了方法有效性。
AI 中文摘要
从轨迹数据中发现动力系统是应用数学与工程学的核心问题。尽管近年机器学习的进展推动了数据驱动系统辨识的发展，但针对不连续动力系统的研究却少得多。这类系统在应用中极具相关性，包括气候动力学与带摩擦的机械系统。本研究考虑直接从轨迹数据中辨识分段光滑动力系统的问题。与光滑情形相比，这需要重构控制方程、检测分隔不同动力学模式的切换超平面，并刻画其行为（如滑动运动）。我们提出了一种模块化框架，先从数据中估计切换超平面，再用几何约束神经网络学习各区域内的光滑动力学。从统计视角研究几何学习阶段，分析不连续性的可辨识性与该方法的鲁棒性。我们还引入了一种具有预设不连续集的新型神经网络架构，并对其逼近性能进行理论分析。该方法在低维基准问题上进行了测试，包括干摩擦振荡器与用于冰期研究的PP04气候模型。
英文摘要
Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.
