Beyond dynamics: learning to discover conservation principles

The discovery of conservation principles is crucial for understanding the fundamental behavior of both classical and quantum physical systems across numerous domains. This paper introduces an innovative method that merges representation learning and topological analysis to explore the topology of conservation law spaces. Notably, the robustness of our approach to noise makes it suitable for complex experimental setups and its aptitude extends to the analysis of quantum systems, as successfully demonstrated in our paper. We exemplify our method's potential to unearth previously unknown conservation principles and endorse interdisciplinary research through a variety of physical simulations. In conclusion, this work emphasizes the significance of data-driven techniques in deepening our comprehension of the principles governing classical and quantum physical systems.