Discovering Multiple Conservation Laws from Trajectories by Machine Learning

Conservation laws are core concepts in dynamical system modeling and the study of physical symmetries. Although machine learning has achieved significant progress in discovering physical laws, existing methods often face challenges such as identifying only a single conserved quantity. To bridge this gap, we introduce OAC-Net. By embedding a gradient orthogonality penalty directly into the neural network’s objective function, OAC-Net enables the simultaneous discovery of multiple independent integrals of motion (IOM) directly from raw trajectory data. Unlike previous heuristic approaches, we prove that gradient orthogonality of real-analytic functions implies the functional independence of the learned integrals of motion. The proposed method is validated on several canonical dynamical systems, including the two-dimensional Kepler system. Experimental results show that OAC-Net efficiently and robustly identifies multiple independent integrals of motion, with the learned integrals of motion exhibiting strong correlations with their true physical values. Additionally, ablation studies confirm OAC-Net’s robustness to hyperparameters such as noise strength and orthogonal penalty coefficient. Our approach provides an effective framework for discovering multiple interpretable integrals of motion from complex trajectory data.