Learning finite symmetry groups of dynamical systems via equivariance detection (2503.03014v1)

Abstract: In this work, we introduce the Equivariance Seeker Model (ESM), a data-driven method for discovering the underlying finite equivariant symmetry group of an arbitrary function. ESM achieves this by optimizing a loss function that balances equivariance preservation with the penalization of redundant solutions, ensuring the complete and accurate identification of all symmetry transformations. We apply this framework specifically to dynamical systems, identifying their symmetry groups directly from observed trajectory data. To demonstrate its versatility, we test ESM on multiple systems in two distinct scenarios: (i) when the governing equations are known theoretically and (ii) when they are unknown, and the equivariance finding relies solely on observed data. The latter case highlights ESM's fully data-driven capability, as it requires no prior knowledge of the system's equations to operate.

Overview of "Learning Finite Symmetry Groups of Dynamical Systems via Equivariance Detection"

The paper introduces the Equivariance Seeker Model (ESM), a novel machine learning framework designed to determine the exact finite symmetry group of dynamical systems. The method focuses on equipping neural networks to discern equivariant transformations, whereby the system's symmetries are mathematically encapsulated, allowing for a systematic approach to uncovering these inherent properties directly from observational data.

Key Contributions

The research leverages the concept of equivariance to detect symmetry groups, which are essential in simplifying the complexity of dynamical systems and in revealing the underlying mathematical structures governing them. The authors propose a loss function that balances the preservation of equivariance with penalties to avoid redundant solutions, ensuring the model identifies all symmetry transformations accurately.

The model is demonstrated to function in two distinct scenarios: when the governing equations of the system are known, and more notably, when only trajectory data is available, emphasizing the method's capability to operate without prior knowledge of the system. This delineation highlights ESM's potential in a fully data-driven context for symmetry discovery.

Experimental Results and Implications

Numerical experiments are conducted on various systems, such as Thomas's symmetric attractor, the Lorenz system, and coupled Duffing oscillators, demonstrating that the proposed method can reliably identify full finite symmetry groups. The rigorous testing on both chaotic and complex dynamical systems illustrates the practical efficacy of ESM in revealing symmetry groups consisting of transformation matrices, even from high-dimensional data.

The paper's results suggest profound implications for theoretical and practical applications in dynamical system analysis and modeling. By automating symmetry discovery, the ESM can contribute to more efficient analysis of physical systems, potentially leading to new insights and methodical simplifications of complex phenomena modeled by differential equations. Apart from offering a new perspective on symmetry detection, this method may stimulate further research into data-driven approaches to explore the fundamental symmetries of a wider array of physical systems.

Future Directions

The paper opens several avenues for future exploration, including enhancing the model's robustness and efficiency with even less data, expanding the ESM to accommodate additional types of symmetry groups, and integrating the model with other numerical and analytical tools to enhance its application scope. Additionally, further exploration could investigate the use of more sophisticated machine learning frameworks or hybrid models to improve predictive accuracy and computational efficiency.

Ultimately, this research marks a significant stride towards harnessing machine learning for fundamental tasks traditionally considered the purview of theoretical physics and applied mathematics, providing a data-centric framework that could be pivotal in the ongoing efforts to decode the intricacies of dynamical systems.