Semi-Supervised Classification of Dynamical Regimes in Hamiltonian Systems Using Poincaré Sections via Contrastive Loss p-Laplacian Propagation

Constantin Ruchkin, Elena Avdyushina

Keywords: dynamical systems, Poincaré section, Graph-based Semi-supervised learning, Label propagation, p-Poisson learning, Contrastive Self-supervised learning, classification.

Abstract:

In this article, we study a recent graph-based semi-supervised machine learning method---Contrastive Loss p-Laplacian Propagation (CLpLP)---for classifying solutions of dynamical systems from two-dimensional Poincar\'e section data. The combination of p-Laplacian label propagation and contrastive learning provides competitive classification accuracy while requiring fewer labeled samples and less training time than classical graph-based approaches.

The proposed method consists of two stages. In the first stage, contrastive learning is used to construct an informative embedding space. In the second stage, labels are propagated on a graph built in that embedding space. The preliminary contrastive training allows for better separation of classes.

This work includes: development of an automated system for analyzing dynamical regimes from Poincar\'e sections; graph discretization of the solution space and formulation of the classification task in a graph-based semi-supervised setting; adaptation of the two-stage CLpLP method to two-dimensional data; and regime classification for a classical mechanical system (rigid-body integrability), where regular and multi-regular orbits are identified, localized, and classified (3 regular cases and 6 multi-regular cases).

To optimize the loss function, we use Newton-based optimization together with a convolutional neural network in the contrastive-learning stage. The method achieves more than 85\% accuracy even with a small number of labels per class, while accounting for accumulation errors and time constraints. This result is sufficient for our broader objective of building an automated real-time system for classifying solutions of dynamical systems.

The proposed approach is designed for scenarios with large collections of observations (trajectories/sections), costly expert labeling, and the need to construct regime maps over wide parameter ranges.

Submission Number: 33