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This research advances machine learning techniques for identifying piecewise-smooth dynamical systems by integrating geometry-constrained neural networks with statistical analysis of discontinuities. The proposed modular framework addresses critical gaps in modeling complex physical phenomena like climate dynamics and friction-based mechanical systems where traditional smooth assumptions fail. By theoretically analyzing the identifiability of switching hyperplanes, this work provides a robust method for recovering governing equations from trajectory data in regimes involving sliding motion.
Learning piecewise-smooth dynamical systems
arXiv:2608.19785v1 Announce Type: cross Abstract: 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.
Abstract: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.
Dynamical Systems (math.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2608.19785 [math.DS]
(or arXiv:2608.19785v1 [math.DS] for this version)
https://doi.org/10.48550/arXiv.2608.19785 Focus to learn more arXiv-issued DOI via DataCite
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