Conservation laws describe physical properties — ‘conserved quantities’ — that do not change over time, playing an essential role in understanding the underlying physics of dynamical systems. Examples include the law of conservation of mass — which states that, in a chemical reaction, mass is neither created nor destroyed — and the law of conservation of energy — which states that the energy of a system cannot be created or destroyed, but rather, it can only be transformed from one form to another. For many complex systems, the conservation laws — and their corresponding conserved quantities — remain unknown. While data-driven approaches have been recently proposed to identify these conservation laws, such methods often require measuring trajectories of the system dynamics with sufficiently low noise and high time resolution, which might not be always available. In a recent work, Marin Soljačić and colleagues introduce an alternative, non-parametric approach — based on manifold learning and optimal transport — that does not require an explicit model or detailed time information to discover conservation laws.
In the proposed approach, the trajectory data of a dynamical system is first collected and normalized. Next, optimal transport is applied to construct a distance matrix containing the distance between each pair of trajectories. Then, a low-dimensional embedding is extracted using diffusion maps — a manifold learning technique — where each point in the embedding corresponds to a distinct set of conserved quantities. Finally, a heuristic score is used to identify the most relevant embedding components, which correspond to the relevant conserved quantities for the system under study. The authors demonstrated the correctness of their method on different physical systems with multiple phases and high-dimensional spatiotemporal dynamics. In addition, they also tested the robustness of their method to noise and partial observations, showing that the results were not impacted by noise or missing information, and compared their approach against prior deep learning-based methods, showing that their manifold learning-based method had the best performance while being about ten times faster. While the proposed approach has yet to be used to uncover previously unknown conserved quantities, it certainly has the potential to be an important tool for analyzing data from complex dynamical systems.
This is a preview of subscription content, access via your institution